Question

The equation of simple harmonic progressive wave is given by Y = a sin 2π  (bt - cx). The maximum particle velocity will be twice the wave velocity if ______.

Options

• c = π a

• c = `1/(2 pi "a")`

• c = `1/(pi"a")`

• c = 2π a

Answer 1

The equation of simple harmonic progressive wave is given by Y = a sin 2π  (bt - cx). The maximum particle velocity will be twice the wave velocity if `underline("c" = 1/(pi"a"))`.

Explanation:

Given, wave equation, y = a sin 2π  (bt - cx)

Compairlng the above equation with the general equation of the progressive wave which is given as,

y = `"A"_0 sin 2pi ("ft" - x/lambda)`

We get, frequency, f = b, wavelength, `lambda = 1/"c"` and amplitude of the wave, A₀ = a

As, we know, that the maximum velocity of the particle,

`"v"_"max" = "A"_0 omega = "a" xx 2pi"b"`   ...(i)

Wave velocity, `"v"_"wave" = "f"lambda`

`=> "v"_"max" = 2"v"_"wave"`

So, by substituting the values from

Eqs. (i) and (ii) in the above relation, we get

`"a"2pi"b" = 2"b"/"c"`

`therefore "c" = 1/("a"pi)`