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[tex]\pi[/tex]kx - 2[tex]\pi[/tex]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[tex]\pi[/tex]kx = 1.5x
=> k = [tex]\frac{0.75}{\pi }[/tex]
=> wave number (k) = [tex]\frac{0.75}{\pi }[/tex]
Also,
=> 50t = 2[tex]\pi[/tex]ft
=> f = frequency
=> f = [tex]\frac{25}{\pi }[/tex]Hz
But,
=> wavelength (λ) = [tex]\frac{2\pi }{k}[/tex] ------- (iii)
Substituting for k = [tex]\frac{0.75}{\pi }[/tex] in equation (iii)
=> λ = (2[tex]\pi[/tex]) ÷ [tex]\frac{0.75}{\pi }[/tex]
=> λ = (2[tex]\pi[/tex]) x [tex]\frac{\pi }{0.75}[/tex] (where [tex]\pi[/tex] = 3.142)
=> λ = 26.33m
(a) To calculate the speed (v) of the wave, we use the formula;
v = f x λ
where f = [tex]\frac{25}{\pi }[/tex] and λ = 26.33
=> v = [tex]\frac{25}{\pi }[/tex] 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.
