Electrons, Waves and Photons: Comprehensive Study Notes on Wave Properties

Fundamentals of Wave Motion and Transfer

Waves are defined by their ability to transfer energy from one location to another without the net transfer of the material or substance through which they travel. A primary classification is the Progessive wave, which is an oscillation that travels through matter or, in specific instances, through a vacuum. While all progessive waves facilitate the movement of energy, the matter of the medium itself remains in its general location.

Taking sound as a specific example of a progessive wave, when an individual hears someone speaking, the energy is carried by vibrations that travel to the listener’s ears. However, the air particles through which the sound travels do not move from the speaker to the listener. In a progessive wave traveling through a medium like air or water, the particles in that medium move from their equilibrium position to a new position. Because these particles exert forces on each other, a displaced particle will influence its neighbors, allowing the wave to propagate.

Classifications of Waves: Transverse and Longitudinal

There are two primary types of waves studied at this stage: transverse waves and longitudinal waves. In a Transverse Wave, the particles of the medium vibrate at an angle of $90^{\circ}$ to the direction of the energy transfer. It is important to explicitly refer to the behavior of the particles to receive credit in descriptions. Transverse waves are characterized by peaks and troughs, representing the points where the oscillating particles are at their maximum displacement from the equilibrium position.

Longitudinal Waves, often referred to as compression waves, involve particles vibrating parallel to the direction of energy transfer. As these waves travel through a medium, they create a series of compressions and rarefactions. Compressions are regions where particles are bunched together, while rarefactions are regions where they are spread apart. Notably, in a longitudinal wave, no single particle travels along the wave; instead, they oscillate back and forth about their fixed equilibrium positions while energy is transferred, typically from left to right.

Practical Examples and Propagation Mnemonics

Transverse waves include water waves on a surface, electromagnetic waves such as light, waves on a stretched spring, and S-waves (secondary waves) produced during an earthquake. To remember the motion of a transverse wave, one can associate the letter T in Transverse with "Top to bottom" motion, signifying the oscillation perpendicular to the direction of travel.

Longitudinal waves include sound waves, P-waves (primary waves) produced in earthquakes, and waves moving through a slinky. A mnemonic for longitudinal waves is the letter L for "Line," indicating that the particles vibrate in the same line or direction as the wave's path. Propagation is defined as the direction in which energy is transferred.

Quantitative Descriptors of Wave Behavior

Several key variables are used to describe wave characteristics. Displacement ($d$) is the distance of a particle from the equilibrium position in a specific direction. Amplitude ($A$) is defined as the maximum displacement from the equilibrium position in a particular direction. The wavelength ($\lambda$) is the minimum distance between two points that oscillate with the same phase, such as the distance between two adjacent crests or two adjacent compressions.

Temporal characteristics include the Time period ($T$), which is the time required for one complete oscillation. Frequency ($f$) is the number of complete oscillations per second, measured in Hertz ($Hz$), where $1\,Hz = 1\,s^{-1}$. The relationship between frequency and time period is expressed by the formula:

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

Wave speed ($v$ or $c$) represents the distance traveled by the wave per unit of time.

The Wave Equation and Derivation

The speed of a wave can be derived using the fundamental relationship of speed equals distance divided by time. A wave travels a distance equal to one wavelength ($\lambda$) in a time interval equal to one period ($T$). Therefore, wave speed can be written as:

$\text{Wave speed} = \frac{\text{Wavelength}}{\text{Period}}$

$\text{Wave speed} = \frac{\lambda}{T}$

Since it is known that $f = \frac{1}{T}$, we can substitute this into the equation to find the standard wave equation:

$v = f \lambda$

In this formula, $v$ is the wave speed measured in $m\,s^{-1}$, $f$ is the frequency in $Hz$, and $\lambda$ is the wavelength in $m$. This equation is typically provided in examinations.

Practical Laboratory Skills: Using an Oscilloscope

An oscilloscope is a vital tool for determining the frequency of a wave. A typical setup involves a signal generator connected to a loudspeaker, with a microphone capturing the sound to feed into the oscilloscope. The oscilloscope screen displays a trace representing a graph of potential difference (p.d.) against time.

The vertical axis is controlled by the Y-gain setting, which determines the voltage represented per division. For instance, if the Y-gain is set to $1.0\,V$, it would be written as $1.0\,V\,div^{-1}$ or sometimes $1.0\,V\,cm^{-1}$ if each square is one centimeter wide. The horizontal axis is the Time base. If each square represents $1.0\,ms$, this is written as $1.0\,ms\,div^{-1}$ or $1.0\,ms\,cm^{-1}$. By measuring the number of divisions for one full cycle and multiplying by the time base, the period ($T$) can be found, which then allows for the calculation of frequency via $f = \frac{1}{T}$.

Wave Intensity and the Inverse Square Law

Intensity ($I$) is formally defined as the radiant power passing through a surface per unit area. Conceptually, it describes how strong or concentrated the energy of a wave is over a given surface. For example, a bright flashlight has high intensity, while a dim flashlight has low intensity. The mathematical formula for intensity is:

$I = \frac{P}{A}$

Where $I$ is Intensity in $W\,m^{-2}$, $P$ is power in $W$, and $A$ is cross-sectional area in $m^2$. In some problems, work done and time may be provided, which can be substituted to find power ($P = \frac{W}{t}$).

When a wave originates from a point source, the power spreads out uniformly in all directions over the surface of a sphere. Because the surface area of a sphere is $A = 4\pi r^2$, the intensity at a distance $r$ from a point source is:

$I = \frac{P}{4\pi r^2}$

This demonstrates an inverse relationship where intensity drops as the distance increases. Furthermore, as the wave spreads, its amplitude decreases. The specific relationship between intensity and amplitude is found to be:

$\text{Intensity} \propto (\text{Amplitude})^2$

Phase and Phase Difference

Different points on a wave are at different stages of their oscillation at any given time, which is referred to as phase. Two particles are considered to be "in phase" if they have the exact same displacement and are moving in the same direction at the same time. Such points are typically separated by a distance of exactly one wavelength.

Phase difference quantifies how much one particle or wave is in front of or behind another. This is expressed as an angle in either degrees or radians. The formula for phase difference is:

$\text{Phase difference} = \frac{x}{\lambda} \times 360^{\circ}$

Where $x$ is the distance between the two points and $\lambda$ is the wavelength. Significant phase states include:

• In-phase: $0^{\circ}$, $360^{\circ}$, or $0$, $2\pi$ radians (separation of 1 wavelength).
• Anti-phase: $180^{\circ}$ or $\pi$ radians (separation of 0.5 wavelengths).

The Principle of Superposition

When two waves pass through the same region of space and overlap, they interact to temporarily form a new wave. After this interaction, the waves continue along their original paths as if nothing had happened. During the interaction, the waves are said to be superposed.

The principle of superposition states that where two waves meet, the sum of their displacements add vectorially. This means the direction of the displacement (positive or negative) must be taken into account.

For example, in constructive superposition (In-phase), if one wave has a displacement of 1 unit and the other has 2 units, the resultant wave displacement is $1 + 2 = 3$. In destructive superposition (Anti-phase), if one wave has a displacement of 2 squares up and the other has 2 squares down, the displacements cancel out: $2 - 2 = 0$.