Question

A wave is described by the expression y(x, t)= (3.0 cm) × cos(1.5x − 50t), where x is in m and t is in s.

Answer 1

Answer:

Speed = 209.5m/s

Direction = Positive x direction

Explanation:

The general wave equation expressing the displacement of a wave travelling in the positive x-direction is written as follows;

y(x, t) = y x cos(2$\pi$kx - 2$\pi$ft)      --------------------(i)

Where;

y(x, t) = the position of the wave at time t

y = the amplitude of the wave,

k = the wave number

f =  frequency of the wave

Given;

y(x, t)= (3.0 cm) × cos(1.5x − 50t)    -------------------(ii)

Comparing equations(i) and (ii)

=> y = amplitude

=> y = 3.0cm

=> y = 0.03m

Also,

=> 2$\pi$kx = 1.5x

=> k = $\frac{0.75}{\pi }$

=> wave number (k) = $\frac{0.75}{\pi }$

Also,

=> 50t = 2$\pi$ft

=> f = frequency

=> f = $\frac{25}{\pi }$Hz

But,

=> wavelength (λ) = $\frac{2\pi }{k}$             ------- (iii)

Substituting for k = $\frac{0.75}{\pi }$ in equation (iii)

=> λ = (2$\pi$) ÷ $\frac{0.75}{\pi }$        

=> λ = (2$\pi$) x $\frac{\pi }{0.75}$        (where $\pi$ = 3.142)

=> λ =  26.33m

(a) To calculate the speed (v) of the wave, we use the formula;

v = f x λ

where f = $\frac{25}{\pi }$ and λ = 26.33

=> v = $\frac{25}{\pi }$ x 26.33

=> v = 209.5m/s

(b) To get the direction in which the wave is travelling, a quick look at the sign between the x and t terms (1.5x - 50t) in the given equation (ii) will suffice.

A negative sign shows that the wave is travelling in the +x direction

A positive sign shows that the wave is travelling in the -x direction.

In this case, the sign between these terms is negative. This shows that the wave is travelling in the positive x direction.