Question

The standing wave on a string is given by the equation y = (0.08 m) sin[120x] cos[150t].

Find

(a) the component waves whose superposition gives this standing wave.

(b) The wavelength and frequency of the standing wave.

(c) The length of the string if 4 nodes and 3 antinodes are formed.

(d) The transverse velocity of the particle of the string at x = 0.04 m and at t = 2 ms.

Answer 1

(a) the component waves whose superposition gives this standing wave:

$$y_s = (0.08 ) \sin[120x] \cos[150t+\pi].$$

(b) The wavelength and frequency of the standing wave:

$$\lambda=2\pi/120=0.052\text{ m},\\ f=150/2\pi=23.9\text{ Hz}.$$

(c) The length of the string if 4 nodes and 3 antinodes are formed:

$$L=\frac32\lambda=0.078\text{ m}.$$

(d) The transverse velocity of the particle of the string at x = 0.04 m and at t = 2 ms.

$$v(x,t) = \frac{∂y(x,t)}{∂t} = -Aω \cos (k x - ω t ),\\\space\\ v(0.04,0.002)=-12\text{ m/s}.$$