31. A simple harmonic wave is expressed by equation y(x, t) = 10 cos (300t -- 0.15x), where y is in cm, x
in m and t in seconds. What is the amplitude and direction of wave propagation? Also calculate the
(i) wavelength (ii) frequency.
Pls answer me
Answers
Explanation:
Answer
(a) The equation of progressive wave travelling from right to left is given by the displacement function:
y(x,t)=asin(ωt+kx+ϕ) ... (i)
The given equation is:
y(x,t)=3.0sin(36t+0.018x+
4
π
) ...(ii)
On comparing both the equations, we find that equation (ii) represents a travelling wave, propgating from right to left.
Now using equations (i) and (ii), we can write:
ω=36 rad/s and k= 0.018 m
−1
We know that:
v=ω/2π and λ=2π/k
Also,
v=fλ
∴v=(ω/2π)×(2π/k)=ω/k
=36/0.018=2000cm/s=20m/s
Hence, the speed of the given travelling wave is 20 m/s.
(b) Amplitude of the given wave, a=3cm
Frequency of the given wave:
f=ω/2π=36/2×3.14=573Hz
(c) On comparing equations (i) and (ii), we find that the intial phase angle, ϕ=π/4
(d) The distance between two successive crests (or troughs) is equal to the wavelength of the wave.
Wavelength is given by the relation: k=2π/λ
∴λ=2π/k=2×3.14/0.018=348.89cm=3.49m