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)