# Waves

### A wave is a disturbance of a medium that transmits energy from one location to another.

wave medium examples
sound matter (any phase) talking, music
light electro-magnetic fields sunlight, wifi, radio
surface interface between two media ocean waves, ripples
string strings, ropes whip, guitar string
seismic solids S and P waves
electrical wires cable TV, phone lines
gravitational spacetime LIGO, Virgo
traffic cars on a road ripples of slow traffic
audience stadium of people The Wave

## Wave Terms

Medium = substance that carries the wave
Propagate = transmit through the medium, how waves move
Equilibrium = resting state of a medium
Amplitude = maximum displacement from equilibrium (m)
Crest = high point on a wave
Trough = low point on a wave

Wavelength, λ = distance between adjacent crests or troughs [m]
Period, T = time for one complete wave cycle to pass a point [s]
Frequency, f = number of cycles that pass per second [Hz, 1/s]

Play around with the settings in the "sine wave" simulation below.
Which of the properties seem related?

amplitude = m           velocity = m/s          wavelength = m
time =
Graphing: Let's see if we can find a relationship between frequency and period. Change the wavelength and record about five different pairs of frequency and period. Graph the pairs with period as the x-coordinate and frequency as the y-coordinate. What does the graph look like?
solution

it's an inverse function

# $$f = \frac{1}{T}$$

$$f$$ = frequency [Hz, 1/s, hertz]
How often an event happens in a second.

$$T$$ = time period [s, seconds]
How many seconds are between each event.
Example: An air horn sounds at a frequency of 220 Hz. How many seconds pass between each wave crest?
solution $$T = \frac{1}{f}$$ $$T = \frac{1}{220}$$ $$T = 0.0045s$$
Interactive: Click the circle to start; click again to pause after it gets annoying.

Think about the definition of period and frequency. Which slider controls period and which controls frequency?
solution

Period is controlled by the left slider and frequency by the right.

Example: The Los Angles Metro Expo Line has a train arrive at 6:50am, 6:56am, and 7:02. What is the period and frequency of the trains?
solution $$6:56-6:50 = 6min$$ $$T = 6 min \left(\frac{60s}{1min}\right) = 360s$$ $$f = \frac{1}{360}$$ $$f=0.0027Hz$$
Example: Red light has a frequency of 450 THz. What is the period of red light? (THz = 10¹²Hz)
solution $$T = \frac{1}{f}$$ $$T = \frac{1}{450 \times 10^{12}}$$ $$T = 2.22 \times 10^{-15}s$$

## Media and Wave Speed

When a wave moves, what is moving is energy, not matter.

The speed of propagation is determined by the medium. Properties like frequency, amplitude, or wavelength generally don't have an effect.

Each type of wave has a different mechanism of propagation. The speed and possibility of a wave propagating through a medium is different for each wave type.

speed (m/s) vacuum air water glass
sound N/A 340 1,484 4,540
light 299,792,458 299,700,000 220,000,000 200,000,000

Sound travels faster in dense media because the atoms are closer together. This means the atoms don't have to move as far to collide.

A light wave moves slower in dense media because as the light wave propagates through a medium it produces ripples that interfere in a way that slows the group velocity of the light wave.

Investigation: Can you figure out what factors affect the speed of a string wave? Go experiment with a string, or slinky to find out.
solution
Tension is probably the easiest way to control the speed of a string wave. The equation below shows that string mass and length are also factors. $$v = \sqrt{\frac{T}{mL}}$$
v = velocity (m/s)
T = tension (N)
m = mass (kg)
L = string length (m)

Changing wave speed is used to change the pitch of stringed instruments. Notes from a guitar or harp are changed through tension, length, or string mass.
Example: Two students are holding a slinky while standing 3.6 m apart. The first student sends a pulse which travels all the way down and back again. It takes 2.4 s for the wave to return. What is the speed of the wave?
solution $$v = \frac{\Delta x}{\Delta t}$$ $$v = \frac{3.6 \times 2}{2.4}$$ $$v = 3 \tfrac{m}{s}$$

## The Wave Equation

For regularly repeating waves the product of the frequency and wavelength is equal to the wave's velocity.

# $$v = f \lambda \quad \quad v = \frac{\lambda}{T}$$

$$v$$ = propagation speed [m/s]
$$\lambda$$ = wavelength [m, meters]
$$f$$ = frequency [Hz, 1/s, hertz]
$$T$$ = time period [s, seconds]
Example: A medium sized air horn sounds at a frequency of 220 Hz. What is the wavelength of the sound wave?
solution

speed of sound at sea level = 340 m/s

$$v = f \lambda$$ $$\lambda = \frac{v}{f}$$ $$\lambda = \frac{340}{220}$$ $$\lambda = 1.54m$$
Example: A cell phone sends and receives light in the microwave range, at wavelengths around 1cm. How many cycles pass through the phone in one second?
solution $$v = 3.0 \times 10^{8} \tfrac{m}{s}$$ $$\lambda = 1cm = 0.01m$$
$$v = f \lambda$$ $$f = \frac{v}{\lambda}$$ $$f = \frac{(3.0 \times 10^{8})}{0.01}$$ $$f=3\times 10^{10}Hz$$ $$f=30 GHz$$
Example: If I triple the wavelength of a sound wave while keeping the wave speed the same, what happens to the frequency? Use proportions.
solution $$\text{v is constant}$$ $$\text{λ and f are inversely proportional}$$ $${ \atop v} {\atop =} { \Downarrow \atop f} { \Uparrow \atop \lambda}$$ $$\text{tripling λ will reduce f by a third}$$
Example: As a wave enters a new medium its speed decreases. The frequency of the wave stays the same, but how does the wavelength change? Use proportions.
solution $$\text{f is constant}$$ $$\text{v and λ are directly proportional}$$ $${\Downarrow \atop v} {\atop =} { \atop f} {\Downarrow \atop \lambda}$$ $$\text{decreasing v will decrease λ}$$
Example: AC (alternating current) electricity in the U.S. has a frequency of 60Hz. In most other countries it is 50Hz. An electrical signal propagates through a wire at about 2/3 the speed of light. What is the wavelength for an AC wave in the United States?
solution $$v = f \lambda$$ $$\lambda = \frac{v}{f}$$ $$\lambda = \frac{(\tfrac{2}{3})(3.0 \times 10^8)}{60}$$ $$\lambda = \frac{2.0 \times 10^8}{60}$$ $$\lambda = 0.033 \times 10^8 m$$ $$\lambda = 3,300km$$
Example: Gravity waves are ripples in spacetime that move at the speed of light. The first detected gravity wave was on 11 February 2016 when the LIGO and Virgo Scientific Collaboration observed gravitational waves originated from a pair of merging black holes. If you convert the gravity wave into a sound wave you can hear a chirp.

How big are gravity waves? Over the 0.2 second detection, the frequency increased from 35Hz to 250Hz. Calculate the starting and ending wavelength.
solution $$v = 3 \times 10^{8} m/s$$ $$v = f \lambda$$ $$\lambda = \frac{v}{f}$$ $$\lambda = \frac{3 \times 10^{8}}{35}$$ $$\lambda = \frac{3 \times 10^{8}}{35}$$ $$\lambda = 8,500,000m$$
$$\lambda = \frac{3 \times 10^{8}}{250}$$ $$\lambda = 1,200,000m$$

The wavelength ranged from 8,500km to 1,200km.

We can also find the length of the entire signal.

$$v = \frac{\Delta x}{\Delta t}$$ $$\Delta x = v\Delta t$$ $$\Delta x = (3 \times 10^{8}) (0.2)$$ $$\Delta x = 0.6 \times 10^{8} m$$

The entire signal was 60,000 km long.

## How Waves Propagate

Propagate is the word we use to describe waves moving. We try not to say move, because waves are a signal transmitting through the medium. The medium doesn't move, just the signal. Each medium has their own mechanism of propagation, but there are some general principles.

A medium at rest is in equilibrium; the forces are in balance. A disruption spreads through the medium bringing it out of equilibrium.

Sound waves are vibrations that propagate through matter. They are produced from changes in pressure and velocity. Your ear senses the amplitude and frequency of the vibrations.

Each time you click the simulations below you will send a pulse of compressed particles, sound!

I've graphed the density and highlighted the particles with higher speeds. You can see the wave trading between kinetic energy and pressure.

Waves don't transmit matter, just energy. In the simulation below one particle is highlighted. Watch how it moves when it becomes part of the wave, but over time it stays in the same area.

## Transverse and Longitudinal

Sound waves are vibrations that move in the same direction the wave moves. Most other waves are produced by vibration perpendicular to the direction the wave moves, like waves on strings

Longitudinal = a wave that vibrates in the direction of propagation