12.1: Transverse Waves
Professional surfer Marcio Freire rides a giant wave at the legendary big wave surf break known as as "Jaws" during one the largest swells of the winter March 13, 2011 in Maui, HI. Massive waves, such as this one, transfer huge amounts of energy.
Transverse Waves
Types of Waves
Water waves, sound waves, and the waves that travel along a rope are mechanical waves. Mechanical waves require a material medium such as water, air, or rope. Light waves, however, are electromagnetic waves and travel without a material medium. They are not mechanical waves.
In all types of mechanical waves, energy moves from one place to another while the media carrying the wave only vibrates back and forth in position. One type of mechanical wave is the transverse wave. In the case of transverse waves, the movement of the medium is perpendicular to the direction of the energy movement.
In the sketch above, consider the transverse wave produced when the boy jerks one end of a rope up and down while the other end is tied to a tree. The energy spent by the boy transfers permanently down the rope to the tree. The rope, however, only moves up and down. If we stuck a piece of tape somewhere on the rope, we would see that the particles of medium do not travel with the energy. After the wave has passed by, the piece of tape would still be in the same place it was before the wave approached. In all transverse waves, the movement media vibrates perpendicularly to the direction of wave motion, and the medium is not permanently moved from one place to another.
Frequency, Wavelength, and Velocity
Waves are identified by several characteristics. There is a center line where the medium would be if there were no wave, which is sometimes describes as the undisturbed position. The displacement of the medium above this undisturbed position is called a crest and the displacement below the undisturbed position is called a trough. The maximums of the crest and trough are equal and are called the amplitude. The distance between equivalent positions on succeeding waves is called the wavelength. The wavelength could be measured from a crest to the next crest or from a trough to the next trough, and is commonly represented with the Greek letter lambda, \begin{align*}\lambda\end{align*}.
The time interval required for one complete wave to pass a point is called the period. During the period of the wave, an entire wavelength from one crest to the next crest passes a position. The number of waves that pass a single position in one second is called the frequency. The period of a wave and its frequency are reciprocals of each other.
\begin{align*}f=\frac{1}{T}\end{align*}
The units for the period are seconds and the units for frequency are s^{-1} or \begin{align*}\frac{1}{\text{s}}\end{align*}. This unit has also been given the name Hertz (Hz).
Another important characteristic of a wave is its velocity. The wave velocity is different from the velocity of the medium; the wave velocity is the velocity of the linearly transferred energy. Since the energy travels one wavelength, \begin{align*}\lambda\end{align*}, in one period, \begin{align*}T\end{align*}, the velocity can be expressed as distance over time:
\begin{align*}v=\frac{\lambda}{T}.\end{align*}
Since period and frequency are reciprocals, the speed of the wave could also be expressed as \begin{align*}v=\lambda f\end{align*}.
Example Problem: A sound wave has a frequency of 262 Hz. What is the time lapse between successive wave crests?
Solution: The time lapse between successive crests would be the period and the period is the reciprocal of the frequency.
\begin{align*}T=\frac{1}{f}=\frac{1}{262 \ \text{s}^{-1}}=0.00382 \ \text{s}\end{align*}
Example Problem: A sound wave has a frequency of 262 Hz has a wavelength of 1.29 m. What is the velocity of the wave?
Solution: \begin{align*}v=\lambda f=(1.29 \ \text{m})(262 \ \text{s}^{-1})=338 \ \text{m/s}\end{align*}
Summary
- Mechanical waves require a material medium such as water, air, or rope.
- In all types of mechanical waves, energy moves from one place to another while the media carrying the wave only vibrates back and forth in position.
- One type of mechanical wave is the transverse wave, in which the movement of the medium is perpendicular to the direction of the energy propagation.
- The maximum displacement of the medium is the distance from the undisturbed position to the top of a crest, or the amplitude.
- The distance along the line of motion of the wave from equivalent positions on succeeding waves is the wavelength.
- The time interval required for one entire wave to pass a point is the period.
- The number of periods per second is the wave's frequency.
- The period of a wave and its frequency are reciprocals of each other.
- The velocity of the wave's energy transfer is given by \begin{align*}v=\lambda f\end{align*} or \begin{align*}v=\frac{\lambda}{T}.\end{align*}
Practice
The following video explains wave characteristics. Pause the video before each practice question and try to solve it yourself before moving on.
http://www.youtube.com/watch?v=5ENLxaPiJJI
- What is the distance between the base line and crest called?
- What symbol is used for wavelength?
- What is the relationship between period and frequency?
Review
- A sound wave produced by a chime 515 m away is heard 1.50 s later.
- What is the speed of sound in air?
- The sound wave has a frequency of 436 Hz. What is its period?
- What is the wavelength of the sound?
- A hiker shouts toward a vertical cliff 685 m away. The echo is heard 4.00 s later.
- What is the speed of sound in air?
- Why is this speed of sound slightly different from the previous answer?
- The wavelength of the sound is 0.750 m. What is the frequency?
- What is the period of the wave?
- The speed of light in air is 3.00 × 10^{8} m/s. If a light wave has a wavelength of 5.80 × 10^{-7} m, what is its frequency?
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Image Attributions
- Describe transverse waves.
- Explain how waves transfer energy without transferring matter.
- Define wavelength, frequency, and period of a transverse wave.
- State the relationship between speed, wavelength, and frequency.
- Solve problems using the relationships between speed, wavelength, frequency, and period.
Concept Nodes:
- transverse wave: A transverse wave is a moving wave that consists of oscillations occurring perpendicular (or right angled) to the direction of energy transfer
- undisturbed position: The equilibrium or rest position of the medium in a wave.
- crest: A crest is the point on a wave with the maximum value or upward displacement within a cycle
- trough: A trough is the opposite of a crest, so the minimum or lowest point in a cycle
- amplitude: The maximum displacement from a zero value during one period of an oscillation.
- wavelength: The distance between corresponding points of two consecutive waves.
- frequency: The number of waves that pass a fixed point in unit time.
- period: A period is the time required for one complete cycle of vibration to pass a given point.