SCIENCE · INVESTIGATE AND EXPLAIN

Audit amplitude and energy at a string junction

A junction can reflect part of a transverse wave even though both strings carry the same tension. The wave speed is √(T/μ), where T is tension and μ is mass per unit length. The supplied characteristic impedance is Z=T/v=√(Tμ), with units kg/s. It describes the traveling-wave transverse force-to-velocity relation; it is not an electrical resistance or an extra loss mechanism.

Learning outcome: Compare density ratios .25,1,4 at fixed tension. Retain signed displacement ratios and positive power fractions separately. Explain the transmitted-amplitude-above-one case using impedance rather than calling it energy amplification.

Starting recap

Separate local oscillation from propagation, amplitude from its square, and a calibration fit from a fresh prediction. Use declared phase, medium frame and boundary conditions.

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Try

At μ2/μ1=4, the impedance ratio is 2. Thus r=−1/3 and t=2/3. Reflected energy is 1/9 and transmitted energy is 2(2/3)²=8/9. Doubling the common tension increases both wave speeds and impedances by √2 but leaves these ratios unchanged.

Open the experiment

Predict

For a second-to-first density ratio of 4 at common tension, what fraction of incident energy is transmitted?

Your prediction and reason

Experiment

Compare density ratios .25,1,4 at fixed tension. Retain signed displacement ratios and positive power fractions separately. Explain the transmitted-amplitude-above-one case using impedance rather than calling it energy amplification.

Check amplitude and energy at a string junction

Worked teaching model · synthetic values · Two ideal semi-infinite strings joined without a point mass, same tension, first density .01 kg/m. Supplied impedance Z=sqrt(T mu), displacement-amplitude r=(Z1−Z2)/(Z1+Z2), t=1+r. Small transverse waves, no dissipation; no real tension apparatus.

Second-to-first linear density ratio: 4 ratio Common tension: 20 N

Reset keeps your prediction, captures and practice history.

Junction amplitude ratios as density changes

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: reflection (solid).2: transmission (dashed). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Junction amplitude ratios as density changes

reflection

second/first density	signed displacement ratio
0.25	0.333333
0.296875	0.294614
0.34375	0.260794
0.390625	0.230769
0.4375	0.203777
0.484375	0.179266
0.53125	0.156826
0.578125	0.136141
0.625	0.116963
0.671875	0.0990945
0.71875	0.0823733
0.765625	0.0666667
0.8125	0.0518633
0.859375	0.0378694
0.90625	0.0246051
0.953125	0.0120017
1	0
1.04688	-0.0114519
1.09375	-0.0223993
1.14063	-0.0328822
1.1875	-0.0429362
1.23438	-0.0525926
1.28125	-0.0618799
1.32813	-0.0708233
1.375	-0.0794457
1.42188	-0.0877677
1.46875	-0.0958082
1.51563	-0.103584
1.5625	-0.111111
1.60938	-0.118403
1.65625	-0.125474
1.70313	-0.132335
1.75	-0.138998
1.79688	-0.145473
1.84375	-0.151769
1.89063	-0.157895
1.9375	-0.163859
1.98438	-0.169669
2.03125	-0.175332
2.07813	-0.180855
2.125	-0.186244
2.17188	-0.191504
2.21875	-0.196641
2.26563	-0.20166
2.3125	-0.206567
2.35938	-0.211364
2.40625	-0.216058
2.45313	-0.220651
2.5	-0.225148
2.54688	-0.229552
2.59375	-0.233867
2.64063	-0.238095
2.6875	-0.24224
2.73438	-0.246305
2.78125	-0.250293
2.82813	-0.254205
2.875	-0.258045
2.92188	-0.261815
2.96875	-0.265518
3.01563	-0.269154
3.0625	-0.272727
3.10938	-0.276239
3.15625	-0.27969
3.20313	-0.283084
3.25	-0.286422
3.29688	-0.289705
3.34375	-0.292935
3.39063	-0.296113
3.4375	-0.299241
3.48438	-0.302321
3.53125	-0.305353
3.57813	-0.30834
3.625	-0.311281
3.67188	-0.314179
3.71875	-0.317035
3.76563	-0.319849
3.8125	-0.322622
3.85938	-0.325357
3.90625	-0.328053
3.95313	-0.330711
4	-0.333333

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

transmission

second/first density	signed displacement ratio
0.25	1.33333
0.296875	1.29461
0.34375	1.26079
0.390625	1.23077
0.4375	1.20378
0.484375	1.17927
0.53125	1.15683
0.578125	1.13614
0.625	1.11696
0.671875	1.09909
0.71875	1.08237
0.765625	1.06667
0.8125	1.05186
0.859375	1.03787
0.90625	1.02461
0.953125	1.012
1	1
1.04688	0.988548
1.09375	0.977601
1.14063	0.967118
1.1875	0.957064
1.23438	0.947407
1.28125	0.93812
1.32813	0.929177
1.375	0.920554
1.42188	0.912232
1.46875	0.904192
1.51563	0.896416
1.5625	0.888889
1.60938	0.881597
1.65625	0.874526
1.70313	0.867665
1.75	0.861002
1.79688	0.854527
1.84375	0.848231
1.89063	0.842105
1.9375	0.836141
1.98438	0.830331
2.03125	0.824668
2.07813	0.819145
2.125	0.813756
2.17188	0.808496
2.21875	0.803359
2.26563	0.79834
2.3125	0.793433
2.35938	0.788636
2.40625	0.783942
2.45313	0.779349
2.5	0.774852
2.54688	0.770448
2.59375	0.766133
2.64063	0.761905
2.6875	0.75776
2.73438	0.753695
2.78125	0.749707
2.82813	0.745795
2.875	0.741955
2.92188	0.738185
2.96875	0.734482
3.01563	0.730846
3.0625	0.727273
3.10938	0.723761
3.15625	0.72031
3.20313	0.716916
3.25	0.713578
3.29688	0.710295
3.34375	0.707065
3.39063	0.703887
3.4375	0.700759
3.48438	0.697679
3.53125	0.694647
3.57813	0.69166
3.625	0.688719
3.67188	0.685821
3.71875	0.682965
3.76563	0.680151
3.8125	0.677378
3.85938	0.674643
3.90625	0.671947
3.95313	0.669289
4	0.666667

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
First wave speed	44.7214	m/s
Second wave speed	22.3607	m/s
First string impedance	0.447214	kg/s
Second string impedance	0.894427	kg/s
Reflected displacement amplitude ratio	-0.333333	ratio
Transmitted displacement amplitude ratio	0.666667	ratio
Reflected energy fraction	0.111111	ratio
Transmitted energy fraction	0.888889	ratio
Energy-fraction residual	-2.22045e-16	ratio

Transmission amplitude may exceed one without creating energy: the transmitted impedance changes too.

Displacement continuity gives t=1+r. The supplied transverse-force boundary relation gives Z1(1−r)=Z2 t, hence the displayed coefficients. For the same frequency, mean traveling-wave power is proportional to Z times amplitude squared, giving R=r² and T=(Z2/Z1)t². Their sum is one. A higher second density gives negative reflection; matched densities give no reflection. A finite-string return pulse or a massive junction would require another model.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

Transmission amplitude may exceed one without creating energy: the transmitted impedance changes too.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Define the observable

Build the bounded model

For a massless junction, displacement continuity gives a transmitted displacement-amplitude ratio t=1+r, where r is the signed reflected ratio. The supplied transverse-force condition is Z1(1−r)=Z2t. Solving these two algebraic equations gives r=(Z1−Z2)/(Z1+Z2) and t=2Z1/(Z1+Z2). Negative r means inversion, not negative reflected energy. These relations are supplied boundary-model enrichment; no calculus derivation is required.

Interpret the evidence

At a common frequency, traveling-wave mean power is proportional to impedance times amplitude squared. Therefore the energy fractions are r² and (Z2/Z1)t², whose sum is one in the ideal lossless junction. Transmission amplitude can exceed the incident amplitude when the second impedance is smaller, yet transmitted power remains at most the incident power. Equal impedances remove reflection. A massive junction, finite-string return pulse, differing boundary tension or dissipation changes the problem; the model does not certify a tensioned physical device.

Worked original case

Predict and investigate

Changed-case practice

Try the changed case before feedback

For μ2/μ1=.25 at equal tension, find the signed reflected displacement-amplitude ratio.

Numerical result in the requested units

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and boundaries

Original prose and cases checked 30 October 2026 against the primary source. These are silent, synthetic wave models: no sound playback, exposure prescription, hearing test or physical apparatus is requested. A correct public number is not independent mastery; retain first explanations and a changed-case review in the course evidence workspace.

Keep the driving frequency across the junction

For this stationary linear junction, reflected and transmitted frequencies equal the driving frequency; their wavelengths change with their medium speeds. Predict/check: at common tension, making the second density four times larger halves its speed, so at unchanged frequency its wavelength halves. It does not double frequency. This is a boundary change, not a Doppler shift caused by moving source or observer.

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Which fresh observation separates a failed physical model from a fitted calibration parameter?

Your explanation and limitations

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