Chapter 15Mechanical Waves

Mechanical Waves

• A disturbance that travels through somematerial or substance known as the medium

• Examples: Water waves, sound

• Not all waves are mechanical, for example,electromagnetic waves

Types of Mechanical Waves

Transverse

Longitudinal

Transverse + Longitudinal

Some Notes on Mechanical Waves

• The disturbance travels or propagates throughthe medium with a definite speed known asthe wave speed

• The medium itself does not travel throughspace, rather, its individual particles undergoback-and-forth or up-and-down motion abouttheir equilibrium

• Waves transport energy, but not matter acrossspace

The Wave Pulse

Periodic Waves

• These are produced when the end of thestring undergoes repetitive, or periodicmotion

• We will consider the case when the endmoves in SHM. The resulting periodic wavethat is produced is known as a sinusoidalwave.

Transverse Sinusoidal Wave

A mass m undergoes SHM with amplitude A, frequency f and period T

Longitudinal Sinusoidal Wave

Wave propagates with speed v & advances onewavelength (λ) in an interval of one period T.Hence,

Constructing the Wave Function

y(x,t) – the wave function

– this tells us the y-position of a particlelocated at x at any time t

Recall that at x = 0, we have SHM:

Pulse at x = 0 moves to the right at a later timet = x/v. Hence, the y position of the maximumshown at time t is equal to the y position of x =0 at an earlier time (t – x/v). In other words,

Hence, we now have

Since cosine is an even function, we can rewrite

But recall that T = 1/f and λ = v/f. Then we canrewrite the wave function further as

Furthermore, define the wave number as k = 2π/λ,then,

For a wave travelling to the negative x direction, it can be shown that

In general,

How do we plot the wave function?

We can chose a specific time, say t = 0 and plot y versus x:

How do we plot the wave function?

Or, we can chose a specific particle, say x = 0 and plot y versus t:

Phase

The argument of the cosine in the wave function(kx ± ωt) is known as the phase.

For a crest (y = A), that is, the cosine function is +1,the phase could have values 0, 2π, 4π, …

For a trough (y = -A), that is, the cosine function is-1, the phase could have values π, 3π, 5π, …

Example

At t = 0, x = 0, a particle has A = 0.075 m, f = 2.00Hz, and v = 12.0 m/s. (a) Find ω, T, λ and k. (b)Write down the wave function y(x,t). (c) Findy(t) at x = 3.0 m.

(a) ω = 12.6 rad/s, T = 0.500 s, λ = 6.00 m, k = 1.05 rad/m

(b) y(x,t) = (0.0750 m)cos[(1.05 rad/m)x-(12.6 m/s)t]

(c) y(t) = -0.075 m cos (12.6 rad/s t)

Problem

A wave function is defined by

Find A, λ, ω, f, v, and the direction of propagation of the wave.