Describing Waves: A Guide to Their Properties
The Anatomy of a Wave
Imagine you are holding one end of a long rope. If you flick your wrist up and down quickly, you create a wave that travels along the rope. This is a transverse wave, where the disturbance is perpendicular to the direction the wave travels. Another type is a longitudinal wave, like sound, where the disturbance is parallel to the direction of travel. To describe any wave, we use a common set of properties that form its unique signature.
| Property | Symbol | Definition | SI Unit | Simple Example |
|---|---|---|---|---|
| Amplitude | A | The maximum displacement of a wave from its rest position. | Meter (m) | The height of an ocean wave from the calm sea level to the crest. |
| Wavelength | λ | The distance between two successive identical points on a wave (e.g., crest to crest). | Meter (m) | The distance between two consecutive wave crests on the ocean. |
| Frequency | f | The number of complete waves (cycles) passing a point per unit of time. | Hertz (Hz) | The number of times a buoy bobs up and down in one second. |
| Period | T | The time taken for one complete wave cycle to pass a point. | Second (s) | The time between one wave crest hitting a pier and the next one hitting it. |
| Wave Speed | v | The distance a wave travels per unit of time. | Meters per second (m/s) | How fast a surfer is propelled forward by an ocean wave. |
Amplitude: The Wave's Energy
The amplitude of a wave is a measure of its maximum disturbance from its rest, or equilibrium, position. Think of it as the wave's "strength" or "intensity." For a transverse wave on a rope, it's the height from the middle of the rope to the top of a crest or the bottom of a trough. In a sound wave, which is longitudinal, the amplitude corresponds to the pressure of the air molecules—higher amplitude means louder sound. In light waves, a greater amplitude means a brighter light. It's crucial to remember that amplitude is related to the energy carried by the wave. A high-energy wave, like the powerful sound from a jet engine, has a large amplitude.
Wavelength, Frequency, and Period: The Wave's Rhythm
These three properties are intimately connected and describe the "rhythm" of the wave.
Wavelength (λ) is the spatial period of the wave—the distance over which the wave's shape repeats. You can measure it from crest to crest, trough to trough, or any two corresponding points.
Frequency (f) and Period (T) are two sides of the same coin. Frequency is the rate at which vibrations occur. If you watch a floating leaf bob up and down as water waves pass by, the frequency is how many times it bobs each second. The period is the duration of one complete bob. They are inversely related by a simple formula:
Formula: Frequency and Period are inverses.
For example, if a wave has a frequency of 2 Hz, it means 2 cycles pass every second. Therefore, the period—the time for one cycle—is 1/2 = 0.5 seconds.
The Universal Wave Equation
Perhaps the most important relationship in wave theory connects the speed of a wave (v) to its frequency (f) and wavelength (λ). This is known as the wave equation.
Formula: The Universal Wave Equation.
Where:
v is wave speed in meters per second (m/s),
f is frequency in Hertz (Hz),
λ is wavelength in meters (m).
This equation tells us that for a wave traveling at a constant speed, frequency and wavelength are inversely proportional. If the frequency increases, the wavelength must decrease, and vice versa. This is why high-frequency sounds (like a whistle) have short wavelengths, and low-frequency sounds (like a bass drum) have long wavelengths.
Waves in Action: From Music to the Ocean
Let's apply these concepts to real-world scenarios to see how they work together.
Example 1: Sound of a Guitar String
When a guitarist plucks a string, it vibrates. The frequency of this vibration determines the musical pitch. A tight, thin string vibrates very quickly, producing a high frequency and thus a high-pitched note. The speed of sound in air is approximately 343 m/s. If the note produced has a frequency of 440 Hz (the standard A note), we can calculate its wavelength in the air:
$ v = f \lambda $
$ 343 = 440 \times \lambda $
$ \lambda = \frac{343}{440} \approx 0.78 \text{ m} $
So, the sound waves traveling through the air from the guitar are about 0.78 meters from one compression to the next.
Example 2: Ocean Waves at the Beach
You're at the beach and notice that a new wave hits the shore every 5 seconds. This is the period, T = 5 s. The frequency is therefore:
$ f = \frac{1}{T} = \frac{1}{5} = 0.2 \text{ Hz} $
You also estimate the distance between waves (wavelength) to be about 20 meters. Using the wave equation, you can find the speed of the waves:
$ v = f \lambda = 0.2 \times 20 = 4 \text{ m/s} $
The ocean waves are traveling at a speed of 4 meters per second.
Common Mistakes and Important Questions
Q: Is the amplitude the same as the distance from a crest to a trough?
A: No, this is a common mistake. The amplitude is the distance from the rest position to a crest (or trough). The distance from a crest to a trough is twice the amplitude. It represents the total vertical displacement between the highest and lowest points of the wave.
Q: If I increase the frequency of a wave, does its speed always increase?
A: Not necessarily. The speed of a wave is usually determined by the medium it's traveling through. For example, the speed of sound is fixed in air at a given temperature. According to $ v = f \lambda $, if the speed (v) is constant and you increase the frequency (f), the wavelength (λ) must decrease to keep the equation balanced. The speed only changes if the wave enters a different medium, like sound going from air to water.
Q: Can two different waves have the same wavelength but different frequencies?
A: Yes, but only if they are traveling at different speeds. From the wave equation $ v = f \lambda $, if the wavelength (λ) is the same for two waves, the one with the higher frequency (f) must have a higher speed (v). For instance, a red light wave and a radio wave could have the same wavelength, but the radio wave travels at the speed of light and has a much lower frequency than the red light wave.
Footnote
1 Hertz (Hz): The SI unit of frequency, defined as one cycle per second.
2 Transverse Wave: A wave in which the particles of the medium move perpendicular to the direction of the wave's advance.
3 Longitudinal Wave: A wave in which the particles of the medium move parallel to the direction of the wave's advance.
4 Equilibrium Position: The resting, or undisturbed, position of a particle in a medium through which a wave is traveling.