Question

Given the wave described by y(x,t)=5cos[π(4x-3t)], in meters. Find the following. Giveexact answers with units.

Answer 1

a) 5 m

b) 0.667 s

c) 0.5 m

d) 0.75 m/s

e) -5 m

Step-by-step explanation:

In an equation of the form

y(x, t) = Acos(kx - ωt)

A is the amplitude, ω = 2π/T where T is the period, and k = 2π/λ where λ is the wavelength. In this case, the equation os

y(x,t) = 5cos(π(4x - 3t)

y(x,t) = 5cos(4πx - 3πt)

So, A = 5, k = 4π, and ω = 3π. Then, we can find each part as follows

a) Amplitude

The amplitude is A, so it is 5 m.

b) the period

Using the equation ω = 2π/T and solving for T, we get:

`T=(2\pi)/(\omega)=(2\pi)/(3\pi)=(2)/(3)=0.667\text{ s}`

So, the period is 0.667 s

c) the wavelength.

using the equation k = 2π/λ and solving for λ, we get:

`\lambda=(2\pi)/(k)=(2\pi)/(4\pi)=0.5\text{ m}`

So, the wavelength is 0.5 m

d) The wave speed

The wave speed can be calculated as the division of the wavelength by the period, so

`v=(\lambda)/(T)=\frac{0.5\text{ m}}{0.667\text{ s}}=0.75\text{ m/s}`

e) The height of the wave at (2, 1)

To find the height, we need to replace (x, t) = (2, 1) on the initial equation, so

`\begin{gathered} y(x,t)=5\cos(\pi(4x-3t)) \\ y(2,1)=5\cos(\pi(4\cdot2-3\cdot1)) \\ y(2,1)=5\cos(\pi(8-3)) \\ y(2,1)=5\cos(\pi(5)) \\ y(2,1)=5\cos(5\pi) \\ y(2,1)=5(-1) \\ y(2,1)=-5 \end{gathered}`

Then, the height of the wave is -5 m.

Therefore, the answers are

a) 5 m

b) 0.667 s

c) 0.5 m

d) 0.75 m/s

e) -5 m