Question

Two waves, each having a frequency of 100 Hz and a wavelength of 2⋅0 cm, are travelling in the same direction on a string. What is the phase difference between the waves (a) if the second wave was produced 0⋅015 s later than the first one at the same place, (b) if the two waves were produced at the same instant but first one was produced a distance 4⋅0 cm behind the second one? (c) If each of the waves has an amplitude of 2⋅0 mm, what would be the amplitudes of the resultant waves in part (a) and (b) ?

Answer 1

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Answer 2

(a). The phase difference is 3π.

(b). The phase difference is 4π.

(c). The resultant amplitude of both waves are 0 and 4 mm.

Explanation:

Given that,

Frequency = 100 Hz

Wavelength = 2.0 cm

Time = 0.015 sec

We need to calculate the speed

Using formula of speed

$v = f\times \lambda$

Put the value into the formula

$v=100\times2\times10^{-2}$

$v=2\ m/s$

(a), We need to calculate the path difference

Using formula of path difference

$x = t\times\lambda$

Put the value into the formula

$x=0.015\times2$

$x=0.03\ m$

We need to calculate the phase difference

$\phi=\dfrac{2\pi x}{\lambda}$

Put the value into the formula

$\phi=\dfrac{2\pi\times 0.03}{2.0\times10^{-2}}$

$\phi=3\pi$

(b). If the two waves were produced at the same instant but first one was produced a distance 4⋅0 cm behind the second one

We need to calculate the phase difference

Using formula of phase difference

$\phi=\dfrac{2\pi x}{\lambda}$

$\phi=\dfrac{2\pi\times4.0\times10^{-2}}{2.0\times10^{-2}}$

$\phi=4\pi\ m$

(c). If each of the waves has an amplitude of 2⋅0 mm,

Let the wave equation for the two waves

$y_{1}=a\sin\omega t$

$y_{2}=a\sin(\omega t+\phi)$

The resultant wave is

$y = y_{1}+y_{2}$

$y=a(\sin\omega t+\sin(\omega t+\phi))$

$y=2a\sin(\omega t+\dfrac{\phi}{2})\cos(\dfrac{\phi}{2})$

Here, resultant amplitude is

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

We need to calculate the amplitude

For $\phi=3\pi\ m$

Put the value into the formula of amplitude

$A=2\times2.0\times10^{-3}\cos(\dfrac{3\pi}{2})$

$A=0$

We need to calculate the amplitude

For $\phi=4\pi\ m$

Put the value into the formula of amplitude

$A=2\times2.0\times10^{-3}\cos(\dfrac{4\pi}{2})$

$A=0.004\ m$

$A=4\ mm$

Hence, (a). The phase difference is 3π.

(b). The phase difference is 4π.

(c). The resultant amplitude of both waves are 0 and 4 mm.

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Topic : phase difference

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