Question

The vertical displacement y(x,t) of a string stretched along the horizontal x-axis is given by y(x,t) = (6.00 mm) cos[(3.25 m-1)x - (7.22 rad/s)t]. (a) What is the minimum time for each complete cycle of the wave? (b) What is the distance between adjacent crests of the wave? (c) How fast does this wave travel?

Answer 1

Answer:

a.0.87sec

b.1.93m

c.2.22m/s

Explanation:

First, let write the general equation describing the equation of a wave and compare parameters.

$y(x,t)=Acos(kx-wt)$

where A is the wave amplitude,

           K is the wave number=$\frac{2\pi }{wavelength}$

           W is the angular frequency =$\frac{2\pi }{period}$

a.Since period is the minimum time for each complete cycle of the wave, we are asked to determine the period

comparing the general equation with the giving wave equation

$y(x,t)=6mmcos(3.25m^{-1}x-7.22rads^{-1} t)$ we can conclude that the angular frequency value is 7.22rad/s

hence $w=\frac{2\pi }{period}\\Period=\frac{2\pi }{w}\\Period=\frac{2\pi }{7.22} \\Period=0.87sec$

Hence the minimum time for each complete cycle of the wave is 0.87sec

b. Since wavelength is define as the distance between adjacent crests or trough of the wave, we are ask to find the wavelength

comparing the general equation with the giving wave equation

$y(x,t)=6mmcos(3.25m^{-1}x-7.22rads^{-1} t)$ we can conclude that the wave number value is 3.25

Hence $K=\frac{2\pi }{wavelength}\\Wavelength=\frac{2\pi }{K} \\Wavelength=\frac{2\pi }{3.25} \\Wavelength=1.93m$.

Hence the distance between adjacent crests of the wave is 1.93m.

c. To determine the speed of the wave we us the equation

$speed=\frac{wavelength}{period}\\ Speed=\frac{1.93}{0.87} \\Speed=2.22m/s$.

Hence the wave travel at a speed of 2.22m/s