Question

A progressive wave travelling along a taut string is given by

Y= 25mm sin 2𝜋 (t/20ms+x/40m)

i. Find the phase velocity of the wave.

ii. Give an equation of a standing wave that can be formed from the 

above progressive wave.

Answer 1

(a) The general equation of the progressive wave can be written as follows:

$$y(x,t)=Asin2\pi(\dfrac{x}{\lambda}+\dfrac{t}{T}),$$

here, $\lambda$ is the wavelength of the wave, $T$ is the period of the wave.

As we can see from the equation of the progressive wave, $\lambda=40\ m, T=20\ s.$

Then, we can find the phase velocity of the wave as follows:

$$v_p=\dfrac{\lambda}{T}=\dfrac{40\ m}{20\ s}=2\ \dfrac{m}{s}.$$

(b) Let's rewrite our equation of progressive wave:

$$y(x,t)=25sin2\pi(\dfrac{x}{40}+\dfrac{t}{20}),$$ $$y(x,t)=25sin(0.05\pi x+0.1\pi t).$$

Let's consider two identical waves that move in opposite directions. The first wave has a wave function of $y_1(x,t)=25sin(0.05\pi x+0.1\pi t)$ and the second wave has a wave function $y_2(x,t)=25sin(0.05\pi x-0.1\pi t)$. The waves interfere and form a resultant wave:

$$y(x,t)=y_1(x,t)+y_2(x,t),$$ $$y(x,t)=25sin(0.05\pi x+0.1\pi t)+25sin(0.05\pi x-0.1\pi t).$$

Using the trigonometric identity

$$sin(\alpha \pm \beta)=sin\alpha cos\beta \pm cos\alpha sin\beta$$

we can write the equation representing the standing wave:

$$y(x,t)=2\cdot25sin(0.05\pi x)cos(0.1\pi t),$$ $$y(x,t)=50sin(0.05\pi x)cos(0.1\pi t).$$