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Question

1. You have an incident wave traveling to the left, 𝑦((𝑥, 𝑡) = −𝐴 ⋅ cos(𝑘𝑥 + 𝜔𝑡), and a reflected wave traveling to the right, 𝑦-(𝑥, 𝑡) = 𝐴 ⋅ cos(𝑘𝑥 − 𝜔𝑡), on a string that is fixed at both ends. The length of the string is 𝐿. Change in sign between the two waves corresponds to a shift in phase of 180°.

a. Find the expression for the standing wave.

b. Show that adjacent nodes are 𝜆/2 apart, where 𝜆 is the wavelength of the standing wave.

c. Find standing wave frequencies.

Answer 1

𝑦1((𝑥, 𝑡) = −𝐴 ⋅ cos(𝑘𝑥 + 𝜔𝑡) .... Eq 1

𝑦2(𝑥, 𝑡) = 𝐴 ⋅ cos(𝑘𝑥 − 𝜔𝑡)........ Eq 2

Using principle of superposition

y = y1+y2

y = −𝐴 ⋅ cos(𝑘𝑥 + 𝜔𝑡) + 𝐴 ⋅ cos(𝑘𝑥 − 𝜔𝑡)

or y= 𝐴 ⋅ cos(𝑘𝑥 − 𝜔𝑡) - 𝐴 ⋅ cos(𝑘𝑥 + 𝜔𝑡)..... Eq 3

Using Cos C - CosD = 2 Sin (C+D)/2 Sin (D-C)/2

y= 2A Sin kx Sin (-wt)

y = - 2A Sin kx *Sin (wt) is the resultant equation of standing wave

Where k = 2∏/𝜆 and w= 2∏/T

For two successive nodes Amplitude of standing wave is zero

To successive values of Sin kx are at kx1= ∏/2 and kx2 =3 ∏/2

Gives k(x2 - x1 )=∏

2∏ (x2-x1) =∏ 𝜆

Thus( x2-x1) =𝜆 /2 which is distance between two successive nodes

The freqency of standing wave is given by w = 2∏ f

or f= w/2∏

(Where ∏ is pi )