Question

Given the wave equation y(x,t) =2.0 Sin (3.0x-4.0t + π/2)m, where x is in meter and t is in second. Find the:

a) amplitude

b)Frequency

c) wavelength

d)speed

e) initial height at x=1.0m

Answer 1

The general equation of wave is:

$$y(x,t) = A\sin(kx-\omega t + \varphi_0)$$

where $k = \dfrac{2\pi}{\lambda}$ is the wave number ($\lambda$ is the wavelength), $\omega = 2\pi f$ is the angular frequency ($f$ is the frequency), $A$ is the amlitude, and $\varphi_0$ is the initial phase.

Comparing the general equation with the given one, find:

$$A = 2m$$ $$\omega = 4rad/s \Rightarrow f =\dfrac{4}{2\pi} \approx 0.64Hz$$ $$k = 3m^{-1}\Rightarrow\lambda = \dfrac{2\pi}{3} \approx 2.09m$$

Speed of the wave is given as:

$$v = f\lambda = 0.64\cdot 2.06m/s \approx 1.33m/s$$

Substituting $x = 1m$, $t = 0$, find:

$$y(1,0) = 2\sin(3\cdot 1 - 4\cdot0 + \pi/2) \approx 0.16m$$

Answer. a) 2m, b) 0.64Hz, c) 2.09m, d) 1.33 m/s, e) 0.16m