A sinusoidal transverse wave travelling on a string is described at t = 2:0 s by

Question

A sinusoidal transverse wave travelling on a string is described at t = 2:0 s by

y(x) = 0.10 sin(4pi x + (33pi/4))

where x and y are in metres. At the point x = 1:0 m in the string, the displacement as a function

of time is given by

y(t) = 0.10 sin((17 pi/4) + 4pi t)

.

a) What are the wavelength and amplitude of the wave?

b) What is the frequency (in Hz) and direction of propagation of the wave?

c) What is the speed of the wave?

d) Write down the full expression for the displacement of the string, y(x; t), at position, x and

time, t.

e) Describe a wave whose simultaneous presence in the string would lead to a standing wave.

Explanation / Answer

a) The first equation gives us the wavelength.

k=2pi/lamda=4pi

lamda=0.5m

amplitude=0.1m (the coefficient of the trigonometric terms)

b)w=4pi=2pi*f

f=2Hz

The direction is towards the negative x axis.

c)v=f*lamda=2*0.5=1m/s

d)y(x,t)=0.1Sin(kx+wt+phi)

y(x,2)=0.10 sin(4pi x + (33pi/4))

y(1,t)=0.10 sin((17 pi/4) + 4pi t)

Comparing,

k=4 pi

w=4pi

phi=17pi/4 -4pi=33pi/4-8pi=pi/4

y(x,t)=0.1Sin(4pi x+4pi t+pi/4)

e)the standing wave will be y(x,t)=0.1Sin(4pi x-4pi t+pi/4)

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