Physics

Physics · optics

Problem

(1) Find the amplitude, frequency, wavelength, and speed of propagation of the wave described by the equation

y=0.2 \cos [\pi(5 t-2 x)]

Here, the units of length and time are taken to be meter and second, respectively. (2) When a sinusoidal wave of amplitude 0.1 m and frequency 2 Hz travels at a speed of

2 \mathrm{~m} / \mathrm{s}

in the

-x

direction, derive the expression for the displacement

y

at position

x

at time

t

by using an integer,

n

. Here, we assume that the displacement at the origin

(x=0)

at time

t=0

is zero, i.e.,

y=0

Intuitive explanation

PART (1)

Wave equations encode all their properties in their coefficients—you just need to match the standard form

y = A\cos(kx - \omega t)

(or its rearrangement) to read off amplitude, angular frequency, and wavenumber, then use the relationships

v = \omega/k

and

\lambda = 2\pi/k

to find the remaining quantities.

PART (2)

A wave traveling in the

-x

direction has the form

y = A\sin(\omega t + kx + \phi)

\cos

with an appropriate phase. The key is using the boundary condition

y(0,0) = 0

to pin down the phase constant, then expressing wavelength in terms of the given speed and frequency.