A wave has the equation (x in meters and t in seconds) y = 0.02sin(30t – 4x). Find its frequency, speed, and length.

Solution,

Given, the equation is y = 0.02 sin(30 t - 4x)

Comparing this equation with the standard wave equation, y = a sin(ωt - kx), where \(k=\frac{2\pi}\lambda\) , we have

ω = 30
Or, 2πf = 30
Or, \(f=\frac{30}{2\pi}\)
Or, \(f=\frac{15}\pi\)
∴ Frequency, f = 4.77 Hz

Also, k = 4 (comparing with the standard wave equation)

Or, \(\frac{2\pi}\lambda=4\)
Or, \(\lambda=\frac{2\pi}4=\frac\pi2\)

∴ Wavelength, λ = 1.571 m

And speed, v = ?

We know that,

Speed, v = λf = \(\frac\pi2\times\frac{15}\pi\) = 7.5 ms^-1 

Hence, frequency, f = 4.77 Hz
Speed, v = 7.5 ms^-1 
and wavelength, λ = 1.571 m

The equation of the wave moving in the opposite direction and double the amplitude is 

y = 0.04 sin (30t + 4x)