The Central Idea
A wave describes a repeating or propagating disturbance distributed through space and time.
A basic one-dimensional traveling wave can be represented by
\[ \boxed{ y(x,t) = A\sin(kx-\omega t+\phi) }. \]This formula combines:
- amplitude \(A\);
- position \(x\);
- time \(t\);
- wave number \(k\);
- angular frequency \(\omega\);
- phase constant \(\phi\).
The spatial pattern and its motion are inseparable parts of the wave model.
Amplitude
Amplitude measures the maximum displacement from equilibrium in a simple wave model.
If
\[ y(x,t)=A\sin(\cdots), \]then the wave ranges between
\[ -A \]and
\[ A. \]Amplitude affects the strength of the disturbance but does not by itself determine wavelength or frequency.
Wavelength
The wavelength \(\lambda\) is the spatial distance between corresponding points on successive cycles.
Examples include:
- crest to crest;
- trough to trough;
- one repeating phase point to the next.
Wavelength is a distance, so it uses length units.
Period
The period \(T\) is the time required for one complete cycle.
If a point on the wave repeats its state every \(0.02\) seconds, then
\[ \boxed{T=0.02\text{ s}}. \]Frequency
The frequency \(f\) is the number of cycles per unit time.
Frequency and period satisfy
\[ \boxed{f=\frac1T} \]and
\[ \boxed{T=\frac1f}. \]Wave Speed
A wave advancing one wavelength during one period has speed
\[ \boxed{v=\frac{\lambda}{T}}. \]Since
\[ f=\frac1T, \]we obtain the fundamental relationship
\[ \boxed{v=f\lambda}. \]Example: Finding Wavelength
A wave travels at
\[ 340\text{ m/s} \]with frequency
\[ 170\text{ Hz}. \]Then
\[ \lambda=\frac{v}{f} = \frac{340}{170} = 2. \]Therefore,
\[ \boxed{\lambda=2\text{ m}}. \]Wave Number
The wave number is
\[ \boxed{k=\frac{2\pi}{\lambda}}. \]It measures spatial phase change per unit distance.
Larger \(k\) corresponds to shorter wavelength.
Angular Frequency
Angular frequency is
\[ \boxed{\omega=2\pi f}. \]Since
\[ f=\frac1T, \]we also have
\[ \boxed{\omega=\frac{2\pi}{T}}. \]Wave Speed from \(k\) and \(\omega\)
Because
\[ k=\frac{2\pi}{\lambda} \]and
\[ \omega=2\pi f, \]the wave speed satisfies
\[ \boxed{v=\frac{\omega}{k}}. \]Phase
Phase identifies where a wave is within its cycle.
Two waves with the same frequency can be shifted relative to one another.
A phase difference affects whether they reinforce or cancel when combined.
Phase Difference from Path Difference
If two waves of wavelength \(\lambda\) travel paths differing by
\[ \Delta L, \]the phase difference is
\[ \boxed{ \Delta\phi = \frac{2\pi}{\lambda}\Delta L }. \]Path geometry therefore directly influences interference.
Constructive Interference
Two waves reinforce when their phases align appropriately.
A common constructive condition is
\[ \boxed{ \Delta L=m\lambda } \]for integer
\[ m. \]The path difference is then a whole number of wavelengths.
Destructive Interference
A common destructive condition is
\[ \boxed{ \Delta L = \left(m+\frac12\right)\lambda }. \]The waves then arrive approximately half a cycle out of phase.
Example: Path Difference
Two coherent waves have wavelength
\[ \lambda=0.50\text{ m}. \]Their path lengths differ by
\[ \Delta L=1.00\text{ m}. \]Then
\[ \frac{\Delta L}{\lambda} = \frac{1.00}{0.50} = 2. \]The path difference is two wavelengths, so it satisfies a constructive interference condition.
Wavefronts
A wavefront is a set of points with the same phase.
In a simplified model:
- a point source can produce circular wavefronts in two dimensions;
- a point source can produce spherical wavefronts in three dimensions;
- a distant source can be approximated locally by parallel plane wavefronts.
Rays
A ray describes a direction of propagation.
In a locally uniform medium, rays are perpendicular to wavefronts.
Ray models are especially useful in geometric optics and simplified acoustic path analysis.
Geometric Optics
Geometric optics models light using rays.
This approach is useful when the wavelength is small relative to the geometric features of interest.
Ray geometry supports:
- reflection;
- refraction;
- mirrors;
- lenses;
- image formation;
- line-of-sight analysis.
The Normal Line
Reflection and refraction angles are measured relative to the normal, a line perpendicular to the surface at the point of interaction.
Confusing the surface itself with the normal changes the angle convention by \(90^\circ\).
Law of Reflection
For ideal specular reflection,
\[ \boxed{\theta_i=\theta_r}, \]where
\[ \theta_i \]is the angle of incidence and
\[ \theta_r \]is the angle of reflection.
Both angles are measured from the surface normal.
Example: Reflection Angle
A ray strikes a mirror at an incidence angle of
\[ 35^\circ \]from the normal.
The reflected ray leaves at
\[ \boxed{35^\circ} \]from the normal on the opposite side.
Reflection by Symmetry
A reflected ray can be constructed geometrically by reflecting the incident direction across the normal line.
The normal is therefore the axis of angular symmetry between the incoming and outgoing rays.
Shortest Broken Path and Reflection
A useful geometric construction reflects one endpoint across a mirror.
The reflected path can then be represented as a straight line to the reflected endpoint.
This explains why equal-angle reflection is connected to shortest-path geometry for a planar mirror.
Refraction
Refraction occurs when a wave changes propagation direction as it passes between media in which its speed differs.
In optics, Snell's law is
\[ \boxed{ n_1\sin\theta_1 = n_2\sin\theta_2 }, \]where \(n_1\) and \(n_2\) are refractive indices.
Refractive Index
In a simplified optical model,
\[ \boxed{ n=\frac{c}{v} }, \]where \(c\) is the speed of light in vacuum and \(v\) is the light speed in the medium.
Larger refractive index corresponds to smaller propagation speed.
Example: Refraction Angle
Suppose
\[ n_1=1.00, \qquad n_2=1.50, \]and
\[ \theta_1=30^\circ. \]Then
\[ 1.00\sin30^\circ = 1.50\sin\theta_2. \]So
\[ \sin\theta_2 = \frac{0.5}{1.5} = \frac13. \]Therefore,
\[ \boxed{ \theta_2 \approx19.5^\circ }. \]Bending Toward and Away from the Normal
In the standard optical model, entering a medium with larger refractive index bends the ray toward the normal.
Entering a medium with smaller refractive index bends it away from the normal, provided transmission occurs.
Critical Angle
When light attempts to pass from a higher-index medium into a lower-index medium, there can be a critical incident angle beyond which no ordinary refracted ray exists.
The critical angle satisfies
\[ \boxed{ \sin\theta_c = \frac{n_2}{n_1} } \]when
\[ n_1>n_2. \]Total Internal Reflection
For incidence angles larger than the critical angle in the applicable geometry, the ray is reflected back into the higher-index medium.
This phenomenon is called total internal reflection.
Thin-Lens Geometry
A simplified thin-lens model relates object distance \(d_o\), image distance \(d_i\), and focal length \(f\):
\[ \boxed{ \frac1f = \frac1{d_o} + \frac1{d_i} }. \]Sign conventions vary, so a specific optics convention must be used consistently.
Magnification
In a common thin-lens convention, linear magnification can be written
\[ \boxed{ m = \frac{h_i}{h_o} = -\frac{d_i}{d_o} }. \]The sign contains orientation information under the adopted convention.
Example: Lens Relationship
Suppose a thin lens has focal length
\[ f=10\text{ cm} \]and an object is located
\[ d_o=30\text{ cm} \]away under a convention where both values are positive.
Then
\[ \frac1{10} = \frac1{30} + \frac1{d_i}. \]Therefore,
\[ \frac1{d_i} = \frac1{10} - \frac1{30} = \frac2{30} = \frac1{15}. \]So
\[ \boxed{d_i=15\text{ cm}}. \]Diffraction
Diffraction is the spreading or bending of waves around obstacles and through openings.
Diffraction becomes especially significant when the relevant opening or obstacle size is comparable to the wavelength.
Pure ray geometry does not capture all diffraction behavior.
Single-Slit Angular Pattern
In a simplified far-field model for a slit of width \(a\), minima occur at
\[ \boxed{ a\sin\theta = m\lambda } \]for nonzero integers
\[ m. \]This relates opening size, wavelength, and angular spreading.
Acoustics
Acoustics studies sound and its propagation.
Spatial acoustic problems involve:
- source position;
- listener position;
- distance;
- reflection;
- absorption;
- interference;
- room geometry;
- travel time.
Sound Travel Time
If sound travels distance \(d\) at speed \(v\), then
\[ \boxed{ t=\frac{d}{v} }. \]Conversely,
\[ \boxed{ d=vt }. \]Example: Sound Delay
Suppose sound travels
\[ 170\text{ m} \]at
\[ 340\text{ m/s}. \]Then
\[ t=\frac{170}{340}=0.5. \]Therefore,
\[ \boxed{t=0.5\text{ s}}. \]Echo Geometry
An echo requires sound to travel from the source to a reflecting surface and then return.
If the source is distance \(d\) from a flat reflecting wall and the sound returns to the source, the total path is approximately
\[ 2d. \]Therefore,
\[ \boxed{ t=\frac{2d}{v} }. \]Finding Distance from Echo Time
Rearranging
\[ t=\frac{2d}{v}, \]gives
\[ \boxed{ d=\frac{vt}{2} }. \]Example: Echo Distance
An echo returns after
\[ 0.40\text{ s} \]when the sound speed is approximated as
\[ 340\text{ m/s}. \]Then
\[ d=\frac{340(0.40)}{2}=68. \]Therefore,
\[ \boxed{d=68\text{ m}}. \]Inverse-Square Spreading
For an ideal point source spreading uniformly through three-dimensional space, intensity decreases with the square of distance:
\[ \boxed{ I\propto\frac1{r^2} }. \]This follows because the same power is distributed over spherical area
\[ 4\pi r^2. \]Intensity from an Ideal Point Source
If a source radiates power \(P\) uniformly in all directions, the idealized intensity at distance \(r\) is
\[ \boxed{ I=\frac{P}{4\pi r^2} }. \]Real sources and environments may differ from this ideal model.
Example: Relative Intensity with Distance
If distance from an ideal point source doubles,
\[ r\longrightarrow2r, \]then
\[ I' \propto \frac1{(2r)^2} = \frac1{4r^2}. \]Therefore, the intensity becomes
\[ \boxed{\frac14} \]of its previous value under the ideal inverse-square model.
Decibels
Sound intensity levels are commonly expressed logarithmically.
A level difference corresponding to intensity ratio
\[ \frac{I_2}{I_1} \]is
\[ \boxed{ \Delta\beta = 10\log_{10} \left( \frac{I_2}{I_1} \right) \text{ dB} }. \]Room Reflections
In a room, sound can reach a listener through:
- a direct path;
- wall reflections;
- ceiling reflections;
- floor reflections;
- multiple-reflection paths.
These paths have different lengths and therefore different arrival times and phases.
Image-Source Construction
A planar acoustic reflection can be modeled by reflecting the source across the reflecting surface.
The reflected path from source to listener then has the same length as a straight path from the image source to the listener.
This converts a broken reflection path into simpler straight-line geometry.
Example: Reflected Path Length
Suppose an image-source construction produces horizontal and vertical separations of
\[ 6\text{ m} \]and
\[ 8\text{ m}. \]The reflected path length is
\[ \sqrt{6^2+8^2} = 10\text{ m}. \]Therefore,
\[ \boxed{10\text{ m}}. \]Standing Waves
When waves traveling in opposite directions interfere, they can form a standing wave.
Standing waves contain:
- nodes, where displacement remains small or zero in the ideal model;
- antinodes, where oscillation amplitude is larger.
String Resonance
For an ideal string fixed at both ends, allowed wavelengths satisfy
\[ \boxed{ L=\frac{n\lambda_n}{2} }, \]where
\[ n=1,2,3,\ldots \]is the mode number.
Therefore,
\[ \boxed{ \lambda_n=\frac{2L}{n} }. \]Resonant Frequencies
Using
\[ v=f\lambda, \]the ideal fixed-string frequencies are
\[ \boxed{ f_n=\frac{nv}{2L} }. \]Example: Fundamental Frequency
An ideal string has length
\[ L=1.0\text{ m} \]and wave speed
\[ v=200\text{ m/s}. \]The fundamental frequency is
\[ f_1=\frac{200}{2(1.0)}=100. \]Therefore,
\[ \boxed{f_1=100\text{ Hz}}. \]Doppler Geometry
Relative motion between a source and observer can change the observed frequency.
The detailed Doppler formula depends on the physical system and motion assumptions.
Geometrically, only the component of relative motion along the line joining source and observer directly changes their separation rate.
Radial Velocity Component
If relative velocity is
\[ \mathbf{v} \]and
\[ \hat{\mathbf{r}} \]is the unit vector along the line of sight, then the radial component is
\[ \boxed{ v_r = \mathbf{v}\cdot\hat{\mathbf{r}} }. \]The dot product extracts the component of motion toward or away from the observer.
Wave Propagation in More Than One Dimension
A wave field may depend on several spatial coordinates:
\[ \boxed{ u=u(x,y,z,t) }. \]Different positions can therefore have different amplitudes and phases at the same time.
This prepares the way for field-based spatial analysis.
Wave Equation
A standard ideal wave equation in three dimensions is
\[ \boxed{ \frac{\partial^2u}{\partial t^2} = c^2\nabla^2u }. \]The operator
\[ \nabla^2 \]measures spatial curvature of the field, while the time derivative measures temporal acceleration.
This equation connects local spatial structure with wave propagation.
Boundaries Matter
Waves interact with boundaries according to the physical properties of the system.
Boundaries may cause:
- reflection;
- transmission;
- absorption;
- refraction;
- diffraction;
- resonance.
Geometry identifies where these interactions occur and how propagation directions change.
Common Error: Confusing Wavelength and Amplitude
Wavelength measures distance between repeating phase points.
Amplitude measures displacement from equilibrium.
They describe different geometric features of a wave.
Common Error: Confusing Frequency and Wave Speed
Frequency measures cycles per second.
Wave speed measures distance traveled per second.
They are related by
\[ v=f\lambda, \]but they are not the same quantity.
Common Error: Measuring Reflection Angles from the Surface
The standard law of reflection measures angles from the normal.
Measuring from the surface instead changes the numerical angle.
Common Error: Forgetting the Return Path in an Echo
Echo time includes travel to the reflector and back.
Therefore,
\[ d=\frac{vt}{2}, \]not
\[ d=vt \]when \(d\) is the one-way distance to the reflecting surface.
Common Error: Using the Inverse-Square Law Without Its Assumptions
The ideal inverse-square model assumes spreading from a point-like source through unobstructed three-dimensional space.
Reflections, directional sources, absorption, boundaries, and confined spaces can produce different behavior.
Common Error: Treating Ray Optics as Universal
Ray models are powerful approximations, but wave effects such as diffraction and interference can become important when wavelength is not negligible compared with geometric feature size.
Common Error: Ignoring Phase in Multiple-Path Problems
Two paths can deliver waves with similar amplitudes but different phases.
Their combination depends on path difference relative to wavelength.
A Reliable Procedure for Basic Wave Problems
- Identify the wave quantity being requested.
- Record known wavelength, frequency, period, speed, or phase information.
- Use consistent units.
- Apply \(v=f\lambda\) or the appropriate periodic relationship.
- For interference, determine path difference.
- For boundary problems, identify the surface normal and medium properties.
- Check whether a ray model or wave model is appropriate.
- Verify that the result is physically and dimensionally reasonable.
A Reliable Procedure for Reflection and Refraction
- Draw the boundary.
- Draw the normal at the interaction point.
- Measure all incidence, reflection, or refraction angles from the normal.
- Use \(\theta_i=\theta_r\) for ideal reflection.
- Use Snell's law for ideal optical refraction.
- Check the refractive-index relationship to predict whether bending should be toward or away from the normal.
A Reliable Procedure for Echo and Acoustic Path Problems
- Identify source, listener, and reflecting surfaces.
- Determine whether the path is direct or reflected.
- Calculate the total path length.
- Use \(t=d/v\) for one-way travel.
- For a same-point echo, remember that total path is twice the one-way distance.
- Compare multiple path lengths when arrival-time or phase differences matter.
Example: Full Wave Workflow
A sound wave travels at
\[ 340\text{ m/s} \]with frequency
\[ 680\text{ Hz}. \]Its wavelength is
\[ \lambda=\frac{340}{680}=0.50\text{ m}. \]Suppose two paths to a listener differ by
\[ 0.25\text{ m}. \]Then
\[ \frac{\Delta L}{\lambda} = \frac{0.25}{0.50} = \frac12. \]The path difference is half a wavelength, corresponding to a phase difference of
\[ \pi \]radians.
Under the simplified coherent equal-frequency model, this is a destructive interference condition.
Practice 1: Wave Speed
A wave has frequency
\[ 50\text{ Hz} \]and wavelength
\[ 4\text{ m}. \]Find its speed.
Practice 2: Period
A wave has frequency
\[ 25\text{ Hz}. \]Find its period.
Practice 3: Wave Number
Find the wave number for
\[ \lambda=2\text{ m}. \]Practice 4: Reflection
A ray strikes a mirror at
\[ 28^\circ \]from the normal. Find the reflection angle.
Practice 5: Refraction
Light passes from a medium with
\[ n_1=1.00 \]into a medium with
\[ n_2=2.00 \]at incidence angle
\[ 30^\circ. \]Find the refracted angle.
Practice 6: Echo Distance
An echo returns after
\[ 0.60\text{ s} \]when sound speed is
\[ 340\text{ m/s}. \]Find the one-way distance to the reflector.
Practice 7: Inverse-Square Change
Under the ideal point-source model, by what factor does intensity change if distance triples?
Practice 8: Constructive or Destructive?
Two coherent waves have wavelength
\[ 0.80\text{ m} \]and path difference
\[ 1.60\text{ m}. \]Does this match a basic constructive or destructive condition?
Practice 9: Standing-Wave Frequency
An ideal string has length
\[ 2\text{ m} \]and wave speed
\[ 120\text{ m/s}. \]Find its fundamental frequency.
Practice 10: Identify the Error
A student measures an incident light ray as \(20^\circ\) above the mirror surface and says the angle of incidence is \(20^\circ\).
Explain the error.
Practice Answers
Practice 1
\[ v=f\lambda = 50(4) = 200. \]Therefore,
\[ \boxed{v=200\text{ m/s}}. \]Practice 2
\[ T=\frac1f = \frac1{25} = 0.04. \]Therefore,
\[ \boxed{T=0.04\text{ s}}. \]Practice 3
\[ k=\frac{2\pi}{2}=\pi. \]Therefore,
\[ \boxed{k=\pi\text{ rad/m}}. \]Practice 4
By the law of reflection,
\[ \theta_r=\theta_i. \]Therefore,
\[ \boxed{28^\circ}. \]Practice 5
\[ 1.00\sin30^\circ = 2.00\sin\theta_2. \] \[ \sin\theta_2 = \frac{0.5}{2} = 0.25. \]Therefore,
\[ \boxed{ \theta_2 \approx14.5^\circ }. \]Practice 6
\[ d=\frac{vt}{2} = \frac{340(0.60)}{2}. \] \[ d=102. \]Therefore,
\[ \boxed{d=102\text{ m}}. \]Practice 7
If
\[ r\longrightarrow3r, \]then
\[ I' = \frac{1}{3^2}I = \frac19I. \]Therefore,
\[ \boxed{\text{intensity becomes }\frac19\text{ as large}}. \]Practice 8
\[ \frac{\Delta L}{\lambda} = \frac{1.60}{0.80} = 2. \]The path difference is an integer multiple of the wavelength.
Therefore,
\[ \boxed{\text{constructive interference}}. \]Practice 9
\[ f_1=\frac{v}{2L} = \frac{120}{2(2)} = 30. \]Therefore,
\[ \boxed{f_1=30\text{ Hz}}. \]Practice 10
Incidence angle is measured from the normal, not from the mirror surface.
Since the normal is perpendicular to the surface,
\[ \theta_i=90^\circ-20^\circ. \]Therefore,
\[ \boxed{\theta_i=70^\circ}. \]What to Remember
- Wavelength describes spatial repetition, while period and frequency describe temporal repetition.
- Wave speed satisfies \(v=f\lambda\).
- Wave number and angular frequency are \(k=2\pi/\lambda\) and \(\omega=2\pi f\).
- Path difference produces phase difference and can create constructive or destructive interference.
- Wavefronts connect points of equal phase, while rays describe local propagation direction.
- The law of reflection uses equal angles measured from the surface normal.
- Snell's law relates refraction angles and refractive indices.
- Ray optics is an approximation and does not capture all diffraction and interference behavior.
- Echo problems use round-trip path length.
- Ideal point-source intensity decreases as \(1/r^2\).
- Room acoustics involves direct and reflected paths with different lengths, delays, and phases.
- Standing waves arise from interference between oppositely traveling waves.
- Resonant frequencies depend on geometry, boundary conditions, and wave speed.
- Relative motion along the line of sight is the key geometric component in Doppler problems.
- Multidimensional wave fields connect spatial curvature with time-dependent propagation.
The central relationship is:
\[ \boxed{ \text{spatial path} + \text{periodicity} + \text{boundaries} \longrightarrow \text{wave behavior} }. \]Where We Go Next
Waves describe disturbances that propagate through spatial systems. Many physical systems are also described by quantities assigned continuously to every point in a region, such as force, velocity, pressure, or temperature.
The next lesson, Fields, Flows, and Physical Systems, develops spatial reasoning for scalar and vector fields, gradients, flow lines, flux, divergence, curl, and physical field behavior.