Match a wave at a boundary before interpreting transmitted power
At a massless join between two strings with the same tension, transverse displacement is continuous and transverse forces balance. For a sinusoidal incident wave, let r and t be signed reflected and transmitted displacement amplitudes divided by incident amplitude. The boundary conditions give 1 + r = t and Z1(1 − r) = Z2t.
Learning outcome: Derive both coefficients from the two boundary equations, retain reflection sign and independently verify the power sum for matched, higher and lower impedances.
Starting recap
Derive equations with their initial or boundary conditions, integrate distributed profiles, and check model domains independently of numerical outputs.
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For a matched second density 0.1 kg/m, Z2 = Z1, r = 0 and t = 1. For second density 0.025 kg/m, Z2 = 1, r = 1/3 and t = 4/3, yet transmitted power fraction is still 8/9.
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For Z1 = 2 and Z2 = 4, the transmitted power fraction is:
Experiment
Derive both coefficients from the two boundary equations, retain reflection sign and independently verify the power sum for matched, higher and lower impedances.
Match displacement and force at a string interface
Worked teaching model · synthetic values · Two semi-infinite ideal strings share tension40 N, with incident density.1 kg/m. A massless joined boundary has continuous displacement and transverse force. Sinusoidal normal traveling waves; no physical strings or sound.
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Displacement reflection versus second density
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: (Z1−Z2)/(Z1+Z2) (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Displacement reflection versus second density
(Z1−Z2)/(Z1+Z2)
| second linear density (kg/m) | signed reflection coefficient |
|---|---|
| 0.025 | 0.3333333 |
| 0.0296875 | 0.2946137 |
| 0.034375 | 0.260794 |
| 0.0390625 | 0.2307692 |
| 0.04375 | 0.2037766 |
| 0.0484375 | 0.1792658 |
| 0.053125 | 0.1568257 |
| 0.0578125 | 0.1361407 |
| 0.0625 | 0.1169631 |
| 0.0671875 | 0.09909446 |
| 0.071875 | 0.08237335 |
| 0.0765625 | 0.06666667 |
| 0.08125 | 0.05186327 |
| 0.0859375 | 0.03786936 |
| 0.090625 | 0.02460505 |
| 0.0953125 | 0.01200173 |
| 0.1 | 0 |
| 0.1046875 | -0.01145188 |
| 0.109375 | -0.02239929 |
| 0.1140625 | -0.03288223 |
| 0.11875 | -0.04293615 |
| 0.1234375 | -0.05259262 |
| 0.128125 | -0.06187988 |
| 0.1328125 | -0.07082327 |
| 0.1375 | -0.07944565 |
| 0.1421875 | -0.0877677 |
| 0.146875 | -0.09580815 |
| 0.1515625 | -0.1035841 |
| 0.15625 | -0.1111111 |
| 0.1609375 | -0.1184035 |
| 0.165625 | -0.1254742 |
| 0.1703125 | -0.1323355 |
| 0.175 | -0.1389983 |
| 0.1796875 | -0.1454728 |
| 0.184375 | -0.1517687 |
| 0.1890625 | -0.1578947 |
| 0.19375 | -0.163859 |
| 0.1984375 | -0.1696692 |
| 0.203125 | -0.1753323 |
| 0.2078125 | -0.1808551 |
| 0.2125 | -0.1862436 |
| 0.2171875 | -0.1915038 |
| 0.221875 | -0.196641 |
| 0.2265625 | -0.2016603 |
| 0.23125 | -0.2065667 |
| 0.2359375 | -0.2113645 |
| 0.240625 | -0.216058 |
| 0.2453125 | -0.2206513 |
| 0.25 | -0.2251482 |
| 0.2546875 | -0.2295523 |
| 0.259375 | -0.2338669 |
| 0.2640625 | -0.2380952 |
| 0.26875 | -0.2422404 |
| 0.2734375 | -0.2463054 |
| 0.278125 | -0.2502928 |
| 0.2828125 | -0.2542053 |
| 0.2875 | -0.2580453 |
| 0.2921875 | -0.2618154 |
| 0.296875 | -0.2655176 |
| 0.3015625 | -0.2691542 |
| 0.30625 | -0.2727273 |
| 0.3109375 | -0.2762387 |
| 0.315625 | -0.2796904 |
| 0.3203125 | -0.2830841 |
| 0.325 | -0.2864217 |
| 0.3296875 | -0.2897046 |
| 0.334375 | -0.2929345 |
| 0.3390625 | -0.296113 |
| 0.34375 | -0.2992413 |
| 0.3484375 | -0.3023211 |
| 0.353125 | -0.3053534 |
| 0.3578125 | -0.3083398 |
| 0.3625 | -0.3112813 |
| 0.3671875 | -0.3141792 |
| 0.371875 | -0.3170346 |
| 0.3765625 | -0.3198486 |
| 0.38125 | -0.3226223 |
| 0.3859375 | -0.3253566 |
| 0.390625 | -0.3280526 |
| 0.3953125 | -0.3307112 |
| 0.4 | -0.3333333 |
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.
| Quantity | Value | Unit |
|---|---|---|
| Incident-string impedance | 2 | kg/s |
| Second-string impedance | 2 | kg/s |
| Reflected displacement coefficient | 0 | ratio |
| Transmitted displacement coefficient | 1 | ratio |
| Reflected displacement amplitude signed | 0 | m |
| Transmitted displacement amplitude | 0.02 | m |
| Reflected power fraction | 0 | ratio |
| Transmitted power fraction | 1 | ratio |
| Power-fraction sum | 1 | ratio |
| Second-string phase speed | 20 | m/s |
Displacement amplitude coefficients are not power fractions when impedances differ.
Continuity gives1+r=t; transverse-force matching gives Z1(1−r)=Z2 t. Thus r=(Z1−Z2)/(Z1+Z2) and t=2Z1/(Z1+Z2). Traveling power at common frequency scales Z A², so transmitted fraction is(Z2/Z1)t². This string displacement convention must not be confused with a pressure-amplitude convention.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
Displacement amplitude coefficients are not power fractions when impedances differ.
Continuity gives1+r=t; transverse-force matching gives Z1(1−r)=Z2 t. Thus r=(Z1−Z2)/(Z1+Z2) and t=2Z1/(Z1+Z2). Traveling power at common frequency scales Z A², so transmitted fraction is(Z2/Z1)t². This string displacement convention must not be confused with a pressure-amplitude convention.
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Why need a displacement coefficient above one not imply excess transmitted power?
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