PHYS 218 Chapter 15

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Mechanical Waves

A disturbance that travels through material

WAVES TRANSPORT ENERGY, NOT MATTER

Several types: (depends on movement of each particle in relation with the movement of the wave)

Transverse Wave

Wave moves along a string when shaken, but each point along the line moves up and down (perpendicular to wave direction)

Longitudinal Wave

Compressing a slinky creates a wave that moves down the spring. Each point inside the slinky moves forward and backward (parallel to wave direction)

Waves can be both transverse and longitudinal in some cases.

Periodic Waves

Simple Harmonic Motion of string/piston/etc.

$\omega =2\pi f={\frac {2\pi }{T}}$

Each particle in material undergoes SHM.

We need a way to relate frequency and wavelength: frequency × wavelength = velocity of wave

$f\lambda =v_{\mathrm {wave} }\,\!$

$v_{\mathrm {wave} }$ depends completely on mechanical properties of the medium and is constant for that medium. In this course, the velocity of the wave does not depend on the frequency

Problem

The speed of sound is known to be 330 m/s.

• Speaker 1 has a frequency of 250 Hz
• Speaker 2 has a frequency of 300 Hz

What is the ratio of wavelength?

${\frac {\lambda _{1}}{\lambda _{2}}}={\frac {v/f_{1}}{v/f_{2}}}={\frac {f_{2}}{f_{1}}}$

Mathematical Description of Waves

A wave moving along the X axis is a 2D function

$y(x,t)$ = displacement from equilibrium position in $y$ component for a point at $x$ along the axis and time $t$

The points $x$ move in a simple harmonic motion as the wave passes.

For $x=0$, we have an equation of simple harmonic motion:

$y(0,t)=A\cos(\omega t+\varphi )$

In general,

${\tfrac {\omega }{v}}={\frac {2\Pi f}{v}}={\frac {2\Pi }{\lambda }}=\kappa$

$\omega =\kappa t$

$y(x,t)=A\cos(\kappa x-\omega t+\varphi )$

Take the derivative to obtain velocity:

$v_{y}(x,t)=-\omega A\sin(\kappa x-\omega t+\varphi )$

Switch sign of ω t for negative $x$ direction

Take second derivative to obtain acceleration:

$a_{y}(x,t)=-\omega ^{2}y(x,t)$

Notice that the second derivative w.r.t. t and x result in $-\omega ^{2}$ and $-\kappa ^{2}$ respectively.

Using $\omega =v\kappa$,

Definition of Wave Equation:

${\frac {\mathrm {d} ^{2}y(x,t)}{\mathrm {d} t^{2}}}=v^{2}{\frac {\mathrm {d} ^{2}y(x,t)}{\mathrm {d} x^{2}}}$

Calculate the Speed of a transverse wave

Limit the system to a small piece $x+\Delta x$

linear mass of a piece times the length of the piece gives the mass of the string in the range

Forces act along length of string: $F_{1}$ and $F_{2}$ at each end of the range

${\begin{aligned}\sum F_{x}:0\\\sum F_{y}:frac{F_{2y}}{F_{2}}={\frac {\mathrm {d} y}{\mathrm {d} x}}{\bigg |}_{x+\Delta x}\end{aligned}}$

${\frac {F_{2y}}{F_{2}}}={\frac {\mathrm {d} y}{\mathrm {d} x}}{\bigg |}_{x+\Delta x}$

…

$v={\sqrt {\frac {F}{\mu }}}$

where F is the tension force and μ is the linear density of the string (mass / length) [kg/m]

Problem

2kg string 80m long with 20kg of tension pulling it. Find velocity of wave and the number of wave cycles in string if frequency is 2Hz

1. $v={\sqrt {\frac {F}{\mu }}}$
2. :(

Forced Oscillations

• Simple Harmonic Motion: $\sum \mathbf {F} =-kx=ma$
• Damped Oscillations: $\sum \mathbf {F} =-kx-b_{v}=ma$ ($b_{v}$ is the damping force)
• Driven Oscillator: $\sum \mathbf {F} =-kx-b_{v}-\underbrace {F_{\mathrm {max} }\cos {\left(\omega _{d}t\right)}} _{\mathrm {oscillator\ force} }=ma$ ($\omega _{d}$ is angular frequency of oscillator)
  • Solution: $x(t)=\underbrace {\frac {F_{\mathrm {max} }}{\sqrt {(k-m\omega _{d})^{2}+b^{2}\omega _{d}^{2}}}} _{\mathrm {amplitude} }\,\cos(\omega _{d}t+\varphi )$
  • What if amplitude is close to ${\sqrt {\tfrac {k}{m}}}$?   $x(t)={\frac {F_{\mathrm {max} }}{b^{2}\omega _{d}^{2}}}\,\cos(\omega _{d}t+\varphi )$

If $b$ is small, then the ${\frac {F_{\mathrm {max} }}{b^{2}}}$ goes to infinity.

Always avoid driving forces close to natural frequency (unless you want to break something; like the earthquake thingy on mythbusters)