Question

Two waves traveling on a string in the same direction both have a frequency of 135 Hz, a wavelength of 2 cm, and an amplitude of 0.04 m. What is the amplitude of the resultant wave if the original waves differ in phase by each of the following values?
(a) p/6 cm(b) p/3 cm

Answer 1

Answer:

The amplitude of the resultant wave are

(a). 0.0772 m

(b). 0.0692 m

Explanation:

Given that,

Frequency = 135 Hz

Wavelength = 2 cm

Amplitude = 0.04 m

We need to calculate the angular frequency

$\omega=2\pi f$

$\omega=2\times\pi\times135$

$\omega=848.23\ rad/s$

As the two waves are identical except in their phase,

The amplitude of the resultant wave is given by

$y+y=A\sin(kx-\omega t)+Asin(kx-\omega t+\phi)$

$y+y=A[2\sin(kx-\omega t+\dfrac{\phi}{2})\cos\phi\dfrac{\phi}{2}$

$y'=2A\cos(\dfrac{\phi}{2})\sin(kx-\omega t+\dfrac{\phi}{2})$

(a). We need to calculate the amplitude of the resultant wave

For $\phi =\dfrac{\pi}{6}$

The amplitude of the resultant wave is

$A'=2A\cos(\dfrac{\phi}{2})$

Put the value into the formula

$A'=2\times0.04\cos(\dfrac{\pi}{12})$

$A'=0.0772\ m$

(b), We need to calculate the amplitude of the resultant wave

For $\phi =\dfrac{\pi}{3}$

$A'=2\times0.04\cos(\dfrac{\pi}{6})$

$A'=0.0692\ m$

Hence, The amplitude of the resultant wave are

(a). 0.0772 m

(b). 0.0692 m