Properties and Representations of Transverse and Longitudinal Waves

Fundamental Properties of Waves

Waves are generated by oscillating sources, and these oscillations travel away from the initial source. Depending on the specific type of wave, oscillations can propagate through a physical medium, such as air or water, or they can travel through a vacuum where no particles are present. To accurately describe the properties of travelling waves, several key terms must be meticulously defined. Wavelength, denoted by the Greek letter $\lambda$ and measured in metres ($\text{m}$), is the distance between a specific point on a wave and the identical point on the subsequent cycle of the wave, such as the distance between two consecutive crests or two consecutive troughs. Amplitude, represented as $\text{A}$ and measured in metres ($\text{m}$), is the magnitude of the maximum displacement reached by an oscillation within the wave from its equilibrium position. Period, denoted by $\text{T}$ and measured in seconds ($\text{s}$), is the time required for one complete oscillation to occur at a single point on the wave. Frequency, denoted by $\text{f}$ and measured in hertz ($\text{Hz}$), refers to the number of complete wave cycles that occur per second. Finally, wave speed, represented by $\text{c}$ or $\text{v}$ and measured in metres per second ($\text{ms}^{-1}$), is the rate at which the wave moves through space. The frequency and the period of a travelling wave are intrinsically related by the equation $\text{f} = \frac{1}{\text{T}}$.

Quantitative Wave Analysis and Worked Examples

In practical applications, these definitions are used to extract data from wave representations. For example, when examining a graph of a travelling wave showing displacement in centimetres against time in milliseconds, one might be asked to determine the amplitude and frequency. If the maximum displacement from the equilibrium position ($\text{x} = 0$) is shown at a specific value, that value is the amplitude. For instance, if a graph shows a peak at $10\,\text{cm}$, the amplitude is converted to metres as $\text{A} = 0.1\,\text{m}$. To calculate frequency, one must first identify the period $\text{T}$ from the graph, which is the time for one complete oscillation. If the period is identified as $1\,\text{ms}$, it must be converted to seconds as $\text{T} = 1 \times 10^{-3}\,\text{s}$. Applying the relationship $\text{f} = \frac{1}{\text{T}}$, the frequency is calculated as $\text{f} = \frac{1}{1 \times 10^{-3}}$, resulting in $\text{f} = 1000\,\text{Hz}$. It is a vital examiner tip to memorise these definitions exactly, as many physics questions begin by requiring a formal definition of these terms.

The Wave Equation and Inverse Relationships

The Wave Equation links wave speed, frequency, and wavelength through the formula $\text{v} = \text{f}\lambda$, where $\text{v}$ is the velocity of the wave in $\text{ms}^{-1}$, $\text{f}$ is the frequency in $\text{Hz}$, and $\lambda$ is the wavelength in $\text{m}$. This equation demonstrates that for a wave travelling at a constant speed, the wavelength and frequency are inversely proportional: as the wavelength increases, the frequency decreases, and conversely, as the wavelength decreases, the frequency increases.

Consider a worked example where a travelling wave has a period of 1.0\,̆s and travels at a velocity of $100\,\text{cm\,s}^{-1}$. To calculate the wavelength in metres, follow these steps: First, identify and convert known quantities: Period \text{T} = 1.0\,̆s = 1.0 \times 10^{-6}\,\text{s} and velocity $\text{c} = 100\,\text{cm\,s}^{-1} = 1.0\,\text{ms}^{-1}$. Second, use the relationship $\text{f} = \frac{1}{\text{T}}$ to find the frequency: $\text{f} = \frac{1}{1 \times 10^{-6}} = 1.0 \times 10^{6}\,\text{Hz}$. Third, rearrange the wave equation $\text{c} = \text{f}\lambda$ to solve for wavelength: $\lambda = \frac{\text{c}}{\text{f}}$. Finally, substitute the values: $\lambda = \frac{1.0}{1 \times 10^{6}} = 1 \times 10^{-6}\,\text{m}$. Students should be prepared to handle various metric prefixes, such as nanometres ($\text{nm} = \text{m} \times 10^{-9}$) or megahertz ($\text{MHz} = \text{Hz} \times 10^{6}$).

Characteristics of Longitudinal Waves

A longitudinal wave is defined as a wave where the particles oscillate parallel to the direction of wave propagation and the direction of energy transfer. These waves are characterised by areas of varying pressure: compressions are regions of high pressure where particles are close together, and rarefactions are regions of low pressure where particles are spread apart. Common examples of longitudinal waves include sound waves, ultrasound waves, and P-waves (primary waves) generated by earthquakes. Crucially, longitudinal waves cannot be polarised. When representing these waves, the wavelength can be measured as the distance between the centres of two consecutive compressions or two consecutive rarefactions. It is important to note that while longitudinal waves involve parallel oscillation, they can still appear sinusoidal when their displacement is plotted against distance on a graph. Therefore, it is essential to read wave descriptions carefully to confirm the direction of oscillation relative to the direction of travel.

Characteristics of Transverse Waves

A transverse wave is defined as a wave where the particles oscillate perpendicular to the direction of wave propagation and the direction of energy transfer. These waves exhibit areas of maximum positive displacement called crests (or peaks) and maximum negative displacement called troughs. Typical examples of transverse waves include all electromagnetic waves (such as radio waves, visible light, and ultraviolet light), vibrations on a guitar string, and waves travelling along a rope or a slinky. Unlike longitudinal waves, transverse waves can be polarised. When defining transverse waves in examinations, students should specify that the vibrations or oscillations are perpendicular to the direction of wave travel. In graphical representations, the distance from the equilibrium line to a crest is the amplitude, and the distance between two consecutive crests or troughs is the wavelength.

Graphical Representation of Waves

Transverse waves are commonly represented using two types of graphs: displacement-distance and displacement-time. Both produce sinusoidal curves but convey different information. A displacement-distance graph shows a "snapshot" of the wave in space at a specific moment; it allows for the direct measurement of amplitude and wavelength ($\lambda$). Upward movement from the centre line is assigned a positive sign, and downward movement is negative. A displacement-time graph shows the movement of a single point on the wave over time; it allows for the measurement of amplitude and the time period ($\text{T}$). From the period, frequency can be calculated as $\text{f} = \frac{1}{\text{T}}$. To determine the future position of a point on a wave, one should sketch the full wave shifted in the direction of travel. Each particle oscillations perpendicular to the direction of travel, staying on its normal line.

Longitudinal waves can also be represented on displacement-distance graphs, which similarly produce sinusoidal shapes. These graphs can be used to identify the location of compressions and rarefactions based on the displacement of particles. Similarly, stationary (standing) waves, which occur when a wave is reflected with a $180^{\circ}$ phase difference, can be transverse or longitudinal. These create a pattern of nodes (points of zero displacement) and antinodes (points of maximum displacement). In a string of length $\text{L}$, a standing wave is formed, and the wavelength $\lambda$ may only represent a portion of the total length $\text{L}$.

Core Practical 6: Investigating the Speed of Sound

The objective of this experiment is to measure the speed of sound in air using an oscilloscope and a signal generator. The independent variable is the distance between the source and the detector, while the dependent variable is the phase of the received signals. Control variables include performing the experiment in the same location and using the same frequency of sound for each set of readings. The required equipment includes a signal generator with a loudspeaker, an oscilloscope with a 2-beam facility, a microphone, two metre rulers (or a $2\,\text{m}$ measuring tape), and connecting leads.

The procedure begins by connecting the microphone and signal generator to the oscilloscope and placing the generator approximately $50\,\text{cm}$ from the microphone. The signal is set to approximately $4\,\text{kHz}$. The oscilloscope time base is adjusted so that approximately three cycles are visible on the screen. The separation between the microphone and speaker is adjusted until a trough on the upper trace (from the generator) coincides with a peak on the lower trace (from the microphone), as this makes alignment easier to judge. This initial distance is recorded as $d_{1}$. The microphone is then moved further away while observing the traces; when the next peak and trough coincide, the new distance $d_{2}$ is recorded.This process is repeated for as many cycles as the space allows. The mean wavelength $\lambda$ is calculated from these measurements, and the frequency $\text{f}$ is determined using the oscilloscope's time base settings ($\text{f} = \frac{1}{\text{T}}$). The process is then repeated with a lower frequency, such as $2\,\text{kHz}$. The speed of sound is calculated using $v = \text{f}\lambda$.

To evaluate the experiment, systematic errors must be considered, particularly ensuring the small scale of the time base (e.g., milliseconds) is accounted for. It is more accurate to find the frequency from the oscilloscope trace rather than the signal generator dial. Random errors are minimised by taking repeat readings and averaging the results. Because sound travels quickly, the distance between instruments should be as large as practical to reduce the relative error in distance measurement. Safety considerations include maintaining sound at normal listening volumes to prevent hearing damage and checking electrical leads for damage. This method is considered highly accurate because the timing is handled automatically by the oscilloscope, eliminating human reaction time as a factor.