Magnetism and magnetic fields Revision Notes

The Wave Speed Equation (v = fλ)

The fundamental relationship between wave speed, frequency, and wavelength is given by the equation: v = fλ.

• v represents the wave speed, measured in metres per second (m/s).
• f represents the frequency, measured in Hertz (Hz).
• λ (lambda) represents the wavelength, measured in metres (m).

This equation applies to all types of waves, both mechanical (e.g., sound waves, water waves) and electromagnetic (e.g., light waves, radio waves). It is essential to use consistent SI units for each variable to obtain correct results in calculations.

Understanding Each Variable

Each variable in the wave speed equation describes a distinct property of a wave:

• Wave Speed (v): This is a measure of how quickly energy is transferred by the wave through a medium. For example, light travels at approximately 3 x 10^8 m/s in a vacuum, while sound travels much slower, around 340 m/s in air.
• Frequency (f): This quantifies the rate at which oscillations occur. A higher frequency means more wave cycles pass a given point per second. It is inversely related to the period (T) of a wave, where T = 1/f.
• Wavelength (λ): This is the spatial characteristic of a wave, representing the distance covered by one complete cycle of the wave. A longer wavelength indicates a more spread-out wave pattern.

Rearranging the Formula for Calculations

You will often need to rearrange the wave speed equation to solve for an unknown quantity. Using the formula v = fλ:

• To find frequency (f), if you know wave speed and wavelength: f = v / λ
• To find wavelength (λ), if you know wave speed and frequency: λ = v / f

Practising these rearrangements ensures you can apply the formula accurately in various problem-solving scenarios. Always double-check that your units are consistent before performing any calculations.

Transverse Waves: Water Ripples as a Model

In a transverse wave, the oscillations of the medium's particles are perpendicular (at 90 degrees) to the direction the wave's energy is travelling. Ripples on water are an excellent model for transverse waves. When an object disturbs the water, water particles move up and down, while the wave itself spreads horizontally. This clearly demonstrates the perpendicular motion characteristic of transverse waves. Other examples include light waves and waves on a stretched string. Key features of water ripples include crests (highest points) and troughs (lowest points).

Measuring the Speed of Water Ripples

To measure the speed of water ripples, you can use a ripple tank or a shallow tray of water. Generate ripples by touching the water surface rhythmically or using a vibrating motor. Measure a known distance (e.g., 1 meter) across the tank. Use a stopwatch to measure the time it takes for a distinct wave crest to travel this distance. Repeat the measurement several times and calculate the average time to increase precision. The speed (v) is then calculated using the formula: v = distance / time. You can also measure wavelength (λ) using a ruler and frequency (f) using the number of waves passing a point per second, then use v = f × λ.

Longitudinal Waves: Sound Waves in Air

In a longitudinal wave, the oscillations of the medium's particles are parallel to the direction of energy transfer. Sound waves travelling through air are the classic example. When a sound source vibrates, it creates regions where air particles are compressed (compressions) and regions where they are spread out (rarefactions). These compressions and rarefactions travel through the air, but the individual air particles only oscillate back and forth in the same direction as the wave's propagation. This parallel motion is the defining feature of longitudinal waves.

Measuring the Speed of Sound in Air

A common method to measure the speed of sound in air involves two people. One person stands a measured distance away from the other (e.g., 100 meters). The first person makes a loud, sharp sound (e.g., by clapping). The second person starts a stopwatch when they *see* the clap and stops it when they *hear* the sound. This eliminates the time taken for light to travel, which is almost instantaneous over this distance. Repeat the experiment multiple times and calculate an average time. The speed of sound is then calculated using the formula: speed = distance / time. This method works best over larger distances to minimise the percentage error in timing.

What is a Wave?

A wave is a disturbance that transfers energy through a medium, or through a vacuum in the case of electromagnetic waves, without transferring matter. Imagine dropping a stone into a pond; the ripples (waves) spread out, but the water itself mostly moves up and down. This demonstrates energy transfer without bulk material movement. Waves are fundamental to how we perceive sound and light, and how many technologies, like radio and mobile phones, operate.

Amplitude and Wavelength

The amplitude of a wave is the maximum displacement of a point on the wave from its undisturbed position. For a transverse wave, this is literally the height of a crest or the depth of a trough from the central line. A greater amplitude indicates that the wave is carrying more energy. The wavelength (λ) is the spatial period of a wave, the distance over which the wave's shape repeats. It is measured from any point on one wave to the identical point on the next wave, for instance, from crest to crest or trough to trough. Wavelength is typically measured in metres (m).

Frequency and Period

The frequency (f) of a wave is the number of complete oscillations or cycles that pass a fixed point per unit of time, usually per second. It is measured in Hertz (Hz), where 1 Hz means one wave cycle per second. The period (T) of a wave is the time taken for one complete wave to pass a fixed point or for a single oscillation to occur. It is measured in seconds (s). Frequency and period are reciprocals of one another, meaning they are inversely related: T = 1/f and f = 1/T. Understanding these relationships is crucial for describing wave behavior.

Waves Transfer Energy, Not Matter

A key characteristic of all waves is their ability to transfer energy from one point to another without transferring matter. The medium through which the wave travels (like water or air) oscillates, or vibrates, around fixed positions. The particles of the medium do not travel along with the wave itself. This distinction is fundamental to understanding how waves work.

Evidence from Water Ripples

To demonstrate that water ripples transfer energy but not water, a simple experiment can be conducted. Place a small, light object (such as a cork or a leaf) on the surface of water. When ripples pass beneath it, the object will be seen to bob up and down, or perhaps move in small circles. Crucially, the object does not travel horizontally along with the ripple. This shows that the water particles themselves are oscillating, primarily vertically, while the wave (and its energy) moves forward.

Evidence from Sound Waves in Air

Sound waves in air provide another clear example of energy transfer without matter transfer. When a sound is produced, such as from a loudspeaker, the air particles near the source are made to vibrate. These vibrating particles collide with neighbouring particles, transferring energy to them. This creates a chain reaction of compressions (areas of higher pressure) and rarefactions (areas of lower pressure) that propagate outwards. However, the individual air molecules do not travel from the loudspeaker to your ear. They simply vibrate back and forth around their equilibrium positions, transmitting the sound energy. If air itself were moving, you would experience a continuous airflow, which does not happen.