Wave Descriptors and the Wave Equation

To describe wave motion mathematically, physicists use specific measurements:

• Amplitude: The maximum displacement of a point on a wave away from its undisturbed position (often called the equilibrium line).
• Wavelength ($\lambda$): The distance from a point on one wave to the exact equivalent point on the adjacent wave.
• Frequency ($f$): The number of waves passing a single point each second, measured in Hertz ($Hz$).
• Period ($T$): The time taken for one complete wave to pass a given point.

Calculating Wave Period

• If you know how many waves arrive every second, you can easily figure out how long each individual wave takes to pass.
• Period and frequency are inversely proportional. As frequency increases, the time period decreases.

The relationship is defined by the period equation:

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

Where:

• $T$ = period in seconds ($s$)
• $f$ = frequency in Hertz ($Hz$)

Worked Example:

A sound wave has a frequency of $2\text{ kHz}$. Calculate the period of the wave.

Step 1: Convert the frequency into standard units ($Hz$).

• $2\text{ kHz} = 2000\text{ Hz}$

Step 2: Substitute the value into the equation.

• $T = \frac{1}{2000}$

Step 3: Calculate the final answer with units.

• $T = 0.0005\text{ s}$ (or $5 \times 10^{-4}\text{ s}$)

The Wave Equation

• All waves, from ocean ripples to the light hitting your eyes, obey one fundamental mathematical rule relating their speed, frequency, and wavelength.
• Wave speed ($v$) is a property of the medium the wave travels through. For example, all electromagnetic waves travel at $3.0 \times 10^8\text{ m/s}$ in a vacuum, while sound travels at approximately $330\text{ m/s}$ in air.

This relationship is calculated using the wave equation:

$$v = f \lambda$$

Where:

• $v$ = wave speed in metres per second ($m/s$)
• $f$ = frequency in Hertz ($Hz$)
• $\lambda$ = wavelength in metres ($m$)

Worked Example:

A radio wave has a wavelength of $2000\text{ m}$ and travels at a speed of $3 \times 10^8\text{ m/s}$. Calculate its frequency.

Step 1: Rearrange the formula to make frequency the subject.

• $f = \frac{v}{\lambda}$

Step 2: Substitute the known values.

• $f = \frac{300,000,000}{2000}$

Step 3: Calculate the final answer with units.

• $f = 150,000\text{ Hz}$ (or $1.5 \times 10^5\text{ Hz}$)

Identifying Waves on Diagrams

• Graphing a wave allows us to 'freeze' it in time or track a single point's motion, revealing its core properties visually.
• On a standard transverse wave diagram, amplitude is the vertical distance from the equilibrium line up to a crest (peak) or down to a trough.
• On a longitudinal wave diagram (like a drawing of a slinky), wavelength must be measured horizontally from the exact center of one compression to the center of the next compression.

When looking at graphs or oscilloscope traces, check the axes carefully:

• Displacement-distance graphs: The horizontal axis represents distance. The horizontal distance between two adjacent crests represents the wavelength.
• Displacement-time graphs: The horizontal axis represents time. The horizontal distance between two adjacent crests represents the period.
• Wavefront diagrams: Viewed from above, each drawn line represents a wavefront (a crest). The gap between two consecutive lines is the wavelength.

Variable Inter-relations (Physics Only)

• Why does sound travel differently underwater compared to in the air?
• When a wave moves from one medium into another, its velocity changes depending on the density and stiffness of the new material.
• However, the frequency remains absolutely constant because it is completely determined by the source that produced the wave.

Because $v = f \lambda$ and frequency is constant across a boundary, velocity and wavelength are directly proportional ($v \propto \lambda$).

• If the wave speeds up in the new medium, its wavelength increases.
• If the wave slows down, its wavelength decreases.
• This change in speed across a boundary is the physical cause of refraction.

Alternatively, if wave speed is kept constant within the same medium, frequency and wavelength are inversely proportional. If a source doubles the frequency of the waves it emits, the wavelength will be perfectly halved.