Two travelling waves of equal amplitudes and equal frequencies move in opposite directions along a string. They interfere to produce a stationary wave whose equation is given by
$\mathrm{y}=\left(10 \cos \pi x \sin \frac{2\pi \mathrm{t}}{\mathrm{T}}\right) \mathrm{cm}$
The amplitude of the particle at $x=\frac{4}{3} \mathrm{cm}$ will be ______ cm.

Complete Solution:

The general form of a stationary wave produced by two waves of equal amplitude and frequency moving in opposite directions is:

y = 2A cos(kx) sin(ωt)

• A is the amplitude of each traveling wave
• k is the wave number
• ω is the angular frequency

In this case, the coefficient 10 cos(π x) represents the amplitude of the stationary wave at position x. Therefore, the amplitude of the wave at any position x is given by 10 cos(π x).

To find the amplitude at x = 4/3 cm, substitute this value into 10 cos(π x):

Amplitude = 10 cos(π × 4/3)

cos(π × 4/3) = cos(4π/3)

The value of cos(4π/3) is -½.

Now we find:

Amplitude = 10 × (-1/2) = -5

Since amplitude is a magnitude, we take the absolute value:

Amplitude = 5 cm

Final Answer:

The amplitude of the particle at x = 4/3 cm is 5 cm.