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.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f \lambda = v_{\mathrm{wave}}\,\!}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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?

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(x,t)} = displacement from equilibrium position in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} component for a point at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} along the axis and time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t}

The points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} move in a simple harmonic motion as the wave passes.

For Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=0} , we have an equation of simple harmonic motion:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(0, t) = A \cos(\omega t + \varphi)}

In general,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{\omega}{v} = \frac{2\Pi f}{v} = \frac{2\Pi}{\lambda} = \kappa}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega = \kappa t}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(x,t) = A \cos(\kappa x - \omega t + \varphi)}


Take the derivative to obtain velocity:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_y(x, t) = -\omega A \sin(\kappa x - \omega t + \varphi)}

Switch sign of ω t for negative Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} direction


Take second derivative to obtain acceleration:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_y(x, t) = -\omega^2 y(x,t)}

Notice that the second derivative w.r.t. t and x result in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\omega^2} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\kappa^2} respectively.

Using Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega = v \kappa} ,

Definition of Wave Equation:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_2} at each end of the range


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \sum F_x: 0 \\ \sum F_y: frac{F_{2y}}{F_2} = \frac{\mathrm{d}y}{\mathrm{d}x} \bigg|_{x+\Delta x} \end{align}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{F_{2y}}{F_2} = \frac{\mathrm{d}y}{\mathrm{d}x} \bigg|_{x+\Delta x}}

…

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v = \sqrt{\frac{F}{\mu}}}
  2. :(


Forced Oscillations

  • Simple Harmonic Motion: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum \mathbf{F} = -kx = ma}
  • Damped Oscillations: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum \mathbf{F} = -kx - b_v = ma} (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_v} is the damping force)
  • Driven Oscillator: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum \mathbf{F} = -kx - b_v - \underbrace{F_\mathrm{max}\cos{\left(\omega_d t\right)}}_{\mathrm{oscillator\ force}} = ma} (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_d} is angular frequency of oscillator)
    • Solution: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{\tfrac{k}{m}}} ?   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x(t) = \frac{F_\mathrm{max}}{b^2 \omega_d^2} \, \cos(\omega_d t + \varphi)}

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} is small, then the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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)

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