Two moving heat sources deposit the same net energy per unit distance. At one observation distance, the faster source gives a higher calculated temperature rise. Farther away, the ordering reverses. Equal line energy has not fixed the entire temperature field.
The short answer is that line energy combines power and travel speed into one ratio, while heat diffusion still responds to speed and distance separately. We will demonstrate the reversal in a declared moving-point-source model, then distinguish a closest-approach reading from a temperature peak and test what the observations can identify.
This guide continues Welding within the Gold Standard of Education and How X Works. All comparisons are original fictional mathematical cases. No values are welding settings, material properties, qualification ranges or predictions of joint performance.
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Name the observation time and distance before comparing equal line-energy sources.
ROUTE 1 · CHAPTERS 1–6
Define the moving heat field
Preserve the energy boundary, coordinates and observation scale.
ROUTE 2 · CHAPTERS 7–13
Compare distance and time
Derive the crossing and distinguish a sampled reading from a peak.
ROUTE 3 · CHAPTERS 14–20
Invert and bound observations
Test special limits, parameter combinations and shared uncertainty.
ROUTE 4 · CHAPTERS 21–28
Preserve physical assumptions
Check reference scales, source geometry, boundaries and changed inputs.
ROUTE 5 · CHAPTERS 29–34
Audit and transfer
Reject incompatible records and choose the next useful observation.
Chapter contents
Define the moving heat field · Chapters 1–6
1. Define net power and line energy on the same boundary
2. State the moving-source model before using its temperature
3. Keep laboratory and moving coordinates distinct
4. Read the temperature field with its dimensions visible
5. Define exactly which observation will be compared
6. Introduce dimensionless quantities without losing the physical question
Compare distance and time · Chapters 7–13
7. Work two equal-line-energy sources at two distances
8. Locate the exact distance where the ordering reverses
9. Explain the competition between power and diffusion
10. Find the speed that maximises this particular reading
11. Check the slow and fast limits carefully
Invert and bound observations · Chapters 14–20
14. Preserve a special centreline result without generalising it
15. Compare late-time behaviour with earlier differences
16. Infer a spatial decay combination from two readings
17. Work an original inverse and a held-out distance
18. Construct indistinguishable closest-approach profiles
19. Check an uncertain speed interval for interior extrema
20. Preserve a common uncertain observation position in a ranking
Preserve physical assumptions · Chapters 21–28
21. Combine amplitude uncertainty only after checking dependence
22. Verify that changing the reference distance changes no physics
23. Keep the diffusion coefficients physically connected
24. Distinguish a point source from a distributed heat footprint
25. Recognise boundaries and losses that invalidate the simple field
26. Separate temperature calculations from residual stress and acceptance
27. Solve a changed equal-line-energy comparison
28. Check the point-source power normalisation independently
Audit and transfer · Chapters 29–34
29. Reject a profile that violates the inverse model's domain
30. Use a time record to break a spatial scaling ambiguity
31. Audit the model with independent physical and numerical checks
32. Choose a learner route with a meaningful changed condition
CHAPTER 1 OF 34 · Define the moving heat field
1. Define net power and line energy on the same boundary
Let P be the constant thermal power entering the modelled material, in watts, and v the source travel speed, in metres per second. Define net line energy ℓ = P/v, in joules per metre.
These quantities already refer to deposited thermal power. They are not an uncorrected electrical voltage-current product. TWI’s heat-input and arc-energy explanation distinguishes those boundaries.
Holding ℓ fixed while increasing v requires P to increase proportionally. Holding P fixed while increasing v is a different comparison because line energy then falls.
Write which comparison is intended before calculating. Otherwise, two correct formulas can appear to disagree simply because one held power constant and the other held energy per distance constant. The numerator’s meaning matters as much as its unit.
CHAPTER 2 OF 34 · Define the moving heat field
2. State the moving-source model before using its temperature
We use an ideal point source moving steadily on the surface of a homogeneous semi-infinite solid. Thermal conductivity k and diffusivity α are positive constants. The surface away from the source is treated as thermally insulating.
The model omits phase change, fluid motion in a molten region, finite source width, finite boundaries and surface heat losses. It represents an established travelling field, without a start or end transient.
This Rosenthal-type idealisation is useful for reasoning about diffusion and travel. It is not a complete welding simulation. In particular, its source singularity prevents treating its central temperature as a real peak.
The primary moving-source literature discusses why distributed sources and additional physics matter. We will use the simple model where its mathematical comparisons are meaningful, then identify precisely which conclusions stop when those assumptions change.
CHAPTER 3 OF 34 · Define the moving heat field
3. Keep laboratory and moving coordinates distinct
Let X be the laboratory coordinate along the travel path. The source position is X = vt. Define x = X − vt, so the source remains at x = 0 in the moving coordinate system.
A point ahead of the source has x > 0. A point behind has x < 0. Let y be the lateral coordinate and z the depth into the ideal solid, with the source on z = 0.
Write R = √(x² + y² + z²). A fixed laboratory sensor does not keep x constant as the source passes; its x decreases at rate v.
This distinction becomes essential when comparing a spatial snapshot with a time history. A point that is closest to the source at one instant may still be warming afterwards because diffusion takes part in the subsequent temperature evolution.
CHAPTER 4 OF 34 · Define the moving heat field
4. Read the temperature field with its dimensions visible
For R > 0, the ideal temperature rise is ΔT = P exp[−v(R + x)/(2α)]/(2πkR). Temperature rise means T − T₀ relative to the specified undisturbed baseline T₀.
The prefactor has units of kelvin: power divided by conductivity times distance. The exponent is dimensionless because vR and α both have units of square metres per second.
Ahead of the source, R + x is large. On the centreline behind it, R + x vanishes. The directional difference is part of the moving field, not a material asymmetry inserted by hand.
The factor 2π belongs to the stated surface-source, semi-infinite geometry. A full-space point-source model has a different geometric prefactor. Moving an equation between those geometries without checking the boundary changes the answer.
CHAPTER 5 OF 34 · Define the moving heat field
5. Define exactly which observation will be compared
Consider a laboratory point at transverse distance b = √(y² + z²) > 0 from the travel axis. At closest approach, x = 0 and R = b.
Its temperature rise at that instant is ΔT_ca = P exp[−vb/(2α)]/(2πkb). Substituting P = ℓv gives ΔT_ca = ℓv exp[−vb/(2α)]/(2πkb).
The subscript ca names closest approach. It does not mean maximum temperature over the sensor’s full history. We will check that difference explicitly later.
This observation definition prevents an apparently minor wording change from creating a false result. Comparing closest-approach readings at two distances is one legitimate mathematical question. Comparing maximum temperatures, cooling times or metallurgical outcomes requires the corresponding separate outputs and assumptions.
CHAPTER 6 OF 34 · Define the moving heat field
6. Introduce dimensionless quantities without losing the physical question
Choose an arbitrary positive reference distance d. Let r = b/d and u = vd/(2α). Define a temperature scale T_* = ℓα/(πkd²).
The closest-approach rise becomes ΔT_ca/T_* = θ = (u/r)exp(−ur). The quantity ur = vb/(2α) compares travel and diffusion on the observation-distance scale.
The reference d is a choice of units for the problem, not an extra physical length in the material. Its role is to let us compare cases without presenting invented material properties or operating speeds as realistic inputs.
All numerical u and r values below belong to this scaled teaching model. Returning to dimensional predictions would require independently supported P, v, k, α, geometry and model validity, not merely a convenient selection of a reference length.
CHAPTER 7 OF 34 · Compare distance and time
7. Work two equal-line-energy sources at two distances
Hold ℓ, k, α and d fixed. Compare u = 1 with u = 3. The second source travels three times faster and supplies three times the deposited power, preserving the same ℓ.
| Scaled distance r | θ for u = 1 | θ for u = 3 | Higher closest-approach reading |
|---|---|---|---|
| 0.50 | 2exp(−0.5) ≈ 1.2131 | 6exp(−1.5) ≈ 1.3388 | Faster source |
| 1.00 | exp(−1) ≈ 0.3679 | 3exp(−3) ≈ 0.1494 | Slower source |
The temperature ordering reverses while line energy stays fixed. The common scale T_* is positive, so it does not change which θ is larger.
No threshold for melting, damage or acceptance has been supplied. The table establishes a comparison between two defined mathematical readings, not a claim that either fictional source produces a better weld.
CHAPTER 8 OF 34 · Compare distance and time
8. Locate the exact distance where the ordering reverses
For speeds v_f > v_s with equal line energy and the same material model, the ratio of closest-approach rises is (v_f/v_s)exp[−(v_f − v_s)b/(2α)].
Setting that ratio to one gives the crossing distance b_c = 2α ln(v_f/v_s)/(v_f − v_s). At smaller positive distances the faster case gives the higher closest-approach reading; at larger distances the slower case does.
For u values 1 and 3, r_c = ln3/2 ≈ 0.5493. The table’s distances 0.50 and 1.00 lie on opposite sides, explaining its reversal.
The crossing is not an empirical material constant. It follows from the chosen speeds, diffusivity, comparison boundary and point-source model. A changed source shape or a different output, such as the full-history peak, requires a new comparison.
CHAPTER 9 OF 34 · Compare distance and time
9. Explain the competition between power and diffusion
At fixed line energy, raising speed increases deposited power. That raises the prefactor in the closest-approach expression. It also increases the exponential suppression at a nonzero transverse distance.
Near enough to the travel axis, the prefactor increase can dominate the comparison. Farther away at the closest-approach instant, the exponential difference can dominate instead.
This does not mean energy disappears in the faster case. We are comparing one time within a spatially and temporally evolving field. Energy deposited earlier and heat arriving later remain part of the broader account.
A useful analogy is observing two journeys at one checkpoint rather than comparing their entire records. The checkpoint can answer its own question accurately while omitting information needed for a different outcome. The mathematical expressions specify exactly which checkpoint is being observed.
CHAPTER 10 OF 34 · Compare distance and time
10. Find the speed that maximises this particular reading
For a fixed physical observation distance b and fixed line energy, differentiate v exp[−vb/(2α)] with respect to v. Its derivative changes sign at v = 2α/b.
In scaled quantities, θ = (u/r)exp(−ur) is maximised at u = 1/r. The maximum is 1/(er²). At r = 0.50, the maximum occurs at u = 2 and equals 4/e ≈ 1.4715.
This is an extremum of a declared mathematical output. It is not an optimum welding speed, because the model and objective omit the actual joint requirements and much of the relevant physics.
The distinction is worth making before interpreting an attractive optimum. A calculus result becomes an engineering recommendation only after its objective, constraints, material description and validation support that use. Here it remains a check on the supplied comparison.
CHAPTER 11 OF 34 · Compare distance and time
11. Check the slow and fast limits carefully
At fixed b > 0 and fixed ℓ, the closest-approach expression tends to zero as v approaches zero. Deposited power ℓv also approaches zero while diffusion has more time to spread the travelling input.
The same expression tends to zero as v increases without bound, because its exponential suppression dominates its linear prefactor at that particular instant and transverse position.
These limits do not say that the entire temperature history vanishes in either case. They concern a closest-approach reading within the ideal established-field model. The high-speed point can reach a later temperature maximum after the source passes.
An asymptotic result must retain its observation definition. Dropping the subscript from ΔT_ca and announcing that high speed eliminates heating would convert a valid restricted limit into an invalid statement about the whole physical process.
CHAPTER 12 OF 34 · Compare distance and time
12. Construct the time history at a fixed laboratory point
Choose a laboratory point with X = 0 and transverse distance b. Let t = 0 be the instant the source passes its longitudinal coordinate. Then x = −vt and R = √(v²t² + b²).
The time history follows by substituting those expressions into the moving-source field. It is not obtained by holding x = 0 throughout time, because that would follow the moving transverse plane rather than the fixed point.
Define scaled time τ = 2αt/d² and ρ = √(u²τ² + r²). The scaled rise is θ(τ) = (u/ρ)exp[−u(ρ − uτ)].
At τ = 0 this reduces to the earlier closest-approach formula. That agreement is an independent consistency check joining the spatial and temporal descriptions without pretending they are the same observation.
CHAPTER 13 OF 34 · Compare distance and time
13. Show that closest approach is not the temperature peak
Differentiate the logarithm of the fixed-point history at τ = 0. Since dρ/dτ is zero there, the result is d lnθ/dτ = u², which is positive for a moving source with u > 0.
Thus, at every nonzero transverse distance in this model, the temperature is still increasing at the closest-approach instant. The peak occurs later, not exactly when x = 0.
For the original u = 1 and u = 3 comparison, the table therefore must not be relabelled as a table of peak temperatures. Establishing a reversal in one sampled output does not automatically establish the same reversal in another.
This is a useful diagnostic habit when reading thermal plots. Ask whether the reported value is at a defined geometrical event, the maximum of a sensor trace, a spatial maximum or a threshold-crossing temperature. Each needs its own extraction rule.
CHAPTER 14 OF 34 · Invert and bound observations
14. Preserve a special centreline result without generalising it
On the centreline behind the moving source, let b = 0 and t > 0 at the fixed laboratory point. Then R = vt and R + x = 0.
The ideal rise reduces exactly to ΔT = ℓ/(2πkt), independent of speed when ℓ and k are held fixed. This is a special result of the ideal centreline, post-passage geometry.
At t = 0 the point-source expression is singular. The formula does not describe a physically measurable maximum at the source, and the model omits the phase change and finite source distribution that matter there.
The special cancellation is still informative. Equal line energy can preserve one modelled time-history quantity while failing to preserve off-axis closest-approach readings. A good explanation keeps both results and names the different conditions under which each was derived.
CHAPTER 15 OF 34 · Invert and bound observations
15. Compare late-time behaviour with earlier differences
At a fixed b > 0 and sufficiently late positive time, R approaches vt to leading order and the exponential factor approaches one. The leading temperature tail is again ℓ/(2πkt).
The two equal-line-energy sources can therefore have different earlier off-axis histories while sharing this leading late-time behaviour in the ideal model. Matching a late tail does not identify the earlier field uniquely.
The statement is asymptotic, not an exact equality at every finite time away from the centreline. Its usefulness depends on whether the omitted higher-order terms are small enough for the intended comparison.
For a paper check, substitute increasingly large scaled τ into the full history and compare θ(τ) with 1/τ. Retain the actual difference. A visually similar tail is evidence about that limit, not permission to replace all preceding temperatures with the same curve.
CHAPTER 16 OF 34 · Invert and bound observations
16. Infer a spatial decay combination from two readings
Assume the baseline temperature is known and both readings are closest-approach rises under one unchanged source and material model. Then bΔT_ca = Aexp(−λb), where A = ℓv/(2πk) and λ = v/(2α).
For distances b₁ < b₂, λ = ln[b₁ΔT₁/(b₂ΔT₂)]/(b₂ − b₁). The amplitude factor A cancels in the ratio.
This identifies a combination of speed and diffusivity. It does not independently separate v from α without additional information. The net line energy and conductivity likewise enter the amplitude through a combined factor.
Use positive rises and compatible observation definitions. A ratio of arbitrary sensor peaks or readings taken at a shared clock time would not satisfy this closest-approach equation merely because the two sensor positions were known.
CHAPTER 17 OF 34 · Invert and bound observations
17. Work an original inverse and a held-out distance
In the scaled model, supply readings θ = 2exp(−0.5) at r = 0.50 and θ = exp(−1) at r = 1. Their weighted ratio is exp(0.5).
Taking its logarithm and dividing by the distance difference 0.50 recovers u = 1. The amplitude rθexp(ur) is then one in this normalisation.
A held-out point at r = 1.50 is predicted to have θ = (2/3)exp(−1.5), about 0.1488. It tests a new observation distance rather than reusing the two points that determined the inverse.
The calculation assumes that the temperature scale used in the supplied numbers is known. With dimensional observations, the ratio still identifies λ, while recovering the separate material and source parameters requires their own measurements or constraints. State which quantities the inverse actually returns.
CHAPTER 18 OF 34 · Invert and bound observations
18. Construct indistinguishable closest-approach profiles
Suppose line energy ℓ is known, while v, α and k are not independently measured. Multiply all three by the same positive factor c.
The ratio v/(2α) is unchanged, and the amplitude ℓv/(2πk) is unchanged. Every closest-approach reading at every transverse distance is therefore identical across this family.
The sources do not have identical deposited power, because P = ℓv changes. Their fixed-point time scales differ as well. An independent travel-speed or deposited-power record can distinguish information that the closest-approach profile alone does not identify.
This example separates a complete profile from a complete parameter determination. More distances can test the functional shape strongly without resolving a parameter ambiguity that preserves the entire profile. The next useful observation must break that invariance rather than merely repeat it more precisely.
CHAPTER 19 OF 34 · Invert and bound observations
19. Check an uncertain speed interval for interior extrema
At r = 0.50, allow u anywhere from 1 to 3 while keeping the common scale fixed. The lowest endpoint gives θ ≈ 1.2131 and the other endpoint gives about 1.3388.
But the maximum occurs at the interior value u = 2, giving 1.4715. An endpoint-only uncertainty calculation would miss that possible higher reading.
The valid range is therefore approximately 1.2131 to 1.4715 under the stated input interval. It is a conditional range, not a statistical confidence interval and not a temperature tolerance for equipment.
The calculation illustrates why monotonicity should be tested before placing uncertainty endpoints into a nonlinear formula. A positive input interval does not imply the output is monotone across it. Here differentiation reveals the missing interior candidate exactly.
CHAPTER 20 OF 34 · Invert and bound observations
20. Preserve a common uncertain observation position in a ranking
Keep u_s = 1 and u_f = 3, but let their common observation distance r lie from 0.50 to 0.60. The fast-to-slow closest-approach ratio is 3exp(−2r).
Its range is approximately 0.9036 to 1.1036. Since this interval crosses one, the supplied position range does not support a guaranteed ordering between the two readings.
The distance must be shared in the comparison. Independently choosing one location to maximise the fast case and another to minimise the slow case would answer a different question from comparing both at the same uncertain position.
The crossing r_c ≈ 0.5493 lies within the allowed interval, so both orderings are genuinely available. No probability distribution over r has been supplied; the calculation cannot say which ordering is more likely.
CHAPTER 21 OF 34 · Preserve physical assumptions
21. Combine amplitude uncertainty only after checking dependence
If line energy alone has an independent positive multiplicative range, the temperature rise scales proportionally at fixed speed, distance and material parameters. That part of the uncertainty is monotone.
For example, multiplying the common temperature scale by a factor from 1 to 1.10 changes the earlier uncertain-speed range to approximately 1.2131 through 1.6187. The maximum uses the larger scale and the interior speed u = 2.
If line energy was calculated from the same uncertain power and speed records, however, it is not automatically independent of u. The original allowed joint records must then be propagated together.
Keeping ℓ fixed in a theoretical comparison and treating an estimated ℓ as independent of measured v are different assumptions. A transparent uncertainty account preserves which one was supplied instead of combining incompatible endpoint choices.
CHAPTER 22 OF 34 · Preserve physical assumptions
22. Verify that changing the reference distance changes no physics
Replace the arbitrary reference d by cd, with c > 0, while keeping all physical quantities fixed. Then u becomes cu, r becomes r/c and T_* becomes T_*/c².
The product ur stays fixed. The scaled expression θ becomes c² times its earlier value, which exactly cancels the changed temperature scale. The dimensional rise T_*θ is unchanged.
This check is useful because dimensionless notation can look like a new physical assumption. It is only a consistent change of representation when every associated scale is transformed together.
A learner who changes d but keeps the old r or temperature scale has changed the calculation inconsistently. The repair is to reconstruct the definitions, not to search for a correction factor after comparing the resulting numbers.
CHAPTER 23 OF 34 · Preserve physical assumptions
23. Keep the diffusion coefficients physically connected
For the ordinary constant-property heat equation, α = k/(ρc_p), where ρc_p is volumetric heat capacity. Conductivity and diffusivity are related through that quantity rather than being unrelated labels.
In the earlier indistinguishable family, scaling k and α together preserves ρc_p. The construction is therefore consistent with that relation while changing both speed and deposited power.
It remains a family of hypothetical materials and source records, not a claim that a particular material can have those properties varied independently in practice. Actual temperature-dependent coefficients would require a different model or justified approximation.
Write the physical parameter relationships before assigning uncertainty intervals. A rectangle of independently varied k and α can imply volumetric heat capacities that were never allowed by the supplied material information. Consistent parameter sets matter in thermal modelling just as consistent observation boundaries do.
CHAPTER 24 OF 34 · Preserve physical assumptions
24. Distinguish a point source from a distributed heat footprint
A point source puts deposited power at an ideal mathematical location. A distributed source spreads it over a finite region. The temperature field then combines contributions from that distribution under the appropriate heat equation and boundaries.
Source width introduces another scale relative to diffusion and travel. Near the source, replacing the distribution with a point can strongly change the calculated field and its peak behaviour.
Eagar and Tsai’s primary work develops travelling distributed-source temperature fields and discusses the limits of the point-source approximation. Its experimental comparisons should not be mistaken for measurements of our invented scaled cases.
The educational lesson is to match model detail to the observation. Farther-field trends may motivate a point approximation, but that is not a universal guarantee. A finite observed footprint, phase change or a required near-source peak is information that can demand a richer description.
CHAPTER 25 OF 34 · Preserve physical assumptions
25. Recognise boundaries and losses that invalidate the simple field
A thin plate, a nearby free edge, a strongly conducting fixture or significant surface heat loss changes the heat-flow problem. The semi-infinite insulated-surface formula should not be silently carried into those cases.
Likewise, multiple passes introduce an initial thermal state and chronology that a single established moving-source field omits. In a linear constant-property model, superposition can help only when its source and boundary assumptions actually apply.
The existing welding heat-input and cooling-rate guide explains the wider recording and timing questions. This article adds the explicit spatial reversal and inverse checks within one declared field.
A model’s failure to represent a boundary is not repaired by more decimal places in its output. The next step is to identify the missing geometry or process information and choose a description that can use it.
CHAPTER 26 OF 34 · Preserve physical assumptions
26. Separate temperature calculations from residual stress and acceptance
A temperature field is one input to a broader thermo-mechanical problem. It does not alone establish distortion, residual stress, microstructure, defect population or service suitability.
The existing residual-stress comparison demonstrates why even a common final dimension can hide different internal mechanical states. Its fictional axial analogy is distinct from our heat-diffusion model.
The ideal fillet-throat guide asks a separate geometry question. Neither calculation converts a thermal model into an acceptance criterion.
For a learner, state the output that has actually been established. A reversed closest-approach temperature comparison is interesting without attaching an unsupported label such as stronger, safer or better to either case.
CHAPTER 27 OF 34 · Preserve physical assumptions
27. Solve a changed equal-line-energy comparison
Compare u_s = 1 and u_f = 2 under the same scaled model. Their closest-approach ratio is 2exp(−r), so the crossing occurs at r = ln2, about 0.6931.
At r = 0.40 the faster-to-slower ratio is about 1.3406. At r = 1 it is about 0.7358. Both observations agree with their locations relative to the crossing.
Now change the question by holding deposited power fixed instead of line energy. The prefactor speed ratio disappears from the closest-approach comparison. For a common P and material model, the faster case has the lower rise at every positive transverse distance.
This is the independent transfer check: changing the held quantity changes the result’s structure. A learner should identify that change before trying to reuse the equal-line-energy crossing formula.
CHAPTER 28 OF 34 · Preserve physical assumptions
28. Check the point-source power normalisation independently
Very close to the ideal source, the exponential tends to one and the leading temperature rise is P/(2πkR). Its leading radial gradient is −P/(2πkR²).
Fourier conduction then carries outward heat flux of magnitude P/(2πR²). A small hemisphere in the solid has area 2πR², so integrating that leading flux gives P. The supplied thermal power and the near-source normalisation agree.
A full sphere would have area 4πR². Using the same temperature prefactor in a full-space problem would count twice the intended source power. The geometry factor is therefore something that can be checked physically, rather than memorised as an unexplained constant.
This limiting check does not make the singular temperature physically realistic. It verifies the source strength represented by the ideal equation. Finite-source temperatures still require a finite-source model.
It also checks a different part of the explanation from the crossing calculation. Two profiles could share a correct relative ranking while both being scaled by an incorrect factor of two. Relative comparisons and absolute power normalisation deserve their own checks.
CHAPTER 29 OF 34 · Audit and transfer
29. Reject a profile that violates the inverse model's domain
For a positive speed and diffusivity, the closest-approach relation requires bΔT to decrease with b. The logarithmic inverse then returns a positive λ.
Suppose two exact fictional rises are both 1 K at scaled distances 0.50 and 1.00, with a known zero baseline and compatible units. Their distance-weighted ratio is 0.50. The inverse returns ln(0.50)/0.50, which is negative.
Those observations do not support the stipulated positive-λ model. Calling the result a negative diffusivity, clipping it to zero or discarding the outer reading would conceal the contradiction.
Real data would require an uncertainty assessment and checks of the baseline, locations, source history and physical assumptions. The exact paper example instead directly rejects this particular closest-approach profile family.
A baseline matters because the formula uses rises T − T₀, not arbitrary absolute thermometer readings. An unknown common offset does not generally cancel inside the weighted ratio. Two temperatures cannot identify an amplitude, decay combination and unknown baseline simultaneously without further information.
For a learner, distinguish an unidentified parameter from an incompatible record. The first allows several supported model states. The second supplies no state within the declared positive-input family.
CHAPTER 30 OF 34 · Audit and transfer
30. Use a time record to break a spatial scaling ambiguity
The earlier family scales v, α and k by a common factor c while preserving ℓ. It leaves every closest-approach profile unchanged. At a fixed laboratory point, it changes the time history in a precise way.
The new distance to the moving source at time t is √(c²v²t² + b²), which equals the original distance evaluated at time ct. The temperature prefactor and exponent therefore give ΔT_new(t) = ΔT_original(ct).
For c = 2, the whole ideal history is compressed by a factor of two in time. Its temperature values and peak height are preserved under that time relabelling, but its peak occurs at half the original elapsed time from passage.
This makes the missing observation concrete. A calibrated time trace can distinguish the cases even when arbitrarily many closest-approach readings cannot. Recording only the maximum height would still lose the time-scale information preserved by the full trace.
The result again belongs to the unchanged point-source and constant-property model. It is not a prescription for varying real material properties. Its purpose is to show why the next measurement should target the parameter combination the earlier observation leaves invariant.
CHAPTER 31 OF 34 · Audit and transfer
31. Audit the model with independent physical and numerical checks
A useful record names the net power boundary, travel direction, baseline, observation coordinates, thermal parameters and point-source assumptions. It distinguishes closest approach, a full-history peak and a late-time approximation.
Check dimensions, the inverse ratio, the crossing equality and the derivative at closest approach. Verify that changing the reference scale leaves dimensional temperature unchanged and that the uncertain-speed interval includes its interior maximum.
The field also satisfies the constant-property travelling heat equation away from its singular source. A numerical derivative check can test that equation at ordinary off-source points, separately from reproducing the closed-form values.
A useful desk exercise hides the source labels until both distance comparisons are complete. Ask the learner to justify each ordering from the common line-energy condition, then reveal the speed ratio. This tests whether the conclusion follows from the field rather than an unsupported intuition that more power must raise every reported temperature.
These checks establish internal consistency of the stated mathematical model. Real-weld interpretation needs relevant measurements, appropriate physical detail and the qualified engineering arrangements discussed in Welding Engineering.
CHAPTER 32 OF 34 · Audit and transfer
32. Choose a learner route with a meaningful changed condition
For repair, work only with power, speed and line-energy units. Explain why equal line energy requires a power change when speed changes.
For stabilisation, reconstruct the scaled two-distance table and derive its crossing. Then compare a closest-approach reading with the fixed-point time history to explain why the table does not report peak temperatures.
For extension, recover the inverse combination, construct the indistinguishable parameter family and propagate the shared-distance uncertainty. Require the learner to identify an observation that would break the remaining ambiguity.
Independent practice should change the held boundary or the information. Hold power fixed, remove the baseline temperature or introduce a finite source footprint. The useful answer begins by explaining which earlier deduction survives and which requires a new model.
CHAPTER 33 OF 34 · Audit and transfer
33. Questions that check the observation and the claim
Does equal line energy determine the same temperature field?
No. The original moving-source comparison preserves line energy while its closest-approach ordering changes with distance.
Is closest approach the same as the temperature peak?
No. The fixed-point derivative is positive at closest approach for the stipulated off-axis model.
Does the calculated speed extremum specify an optimal weld?
No. It optimises one restricted mathematical reading, without the requirements or physical detail needed for that engineering decision.
Can a complete closest-approach profile identify every parameter?
No. The constructed speed–diffusivity–conductivity scaling preserves the entire profile.
Does a correct heat calculation establish joint integrity?
No. Geometry, material response, process evidence and acceptance require their own supported analyses.
CHAPTER 34 OF 34 · Audit and transfer
34. Follow the verified subject and technical sources
Return to Welding for process, competence and evidence, and to the heat-input and cooling-rate guide for the measured energy and timing account.
Primary technical context, checked 26 October 2026, includes Eagar and Tsai, Temperature Fields Produced by Traveling Distributed Heat Sources, published in December 1983, and TWI’s distinction between heat input and arc energy. The first supplies the moving-source framework and approximation limits; the second clarifies the energy boundary. Our scaled comparisons, inverse families, uncertainty bounds and transfer exercises are original derivations, not reported weld measurements or source figures.
Continue through the Education parent and How X Works when another subject asks whether one energy total preserves the spatial and temporal process behind it.