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Waves and Oscillations

Spherical Waves

Spherical waves are three-dimensional mechanical or electromagnetic waves where wavefronts form concentric spheres originating from a point source. Their amplitude decreases inversely with distance from the source to conserve total radiated energy.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Definition of spherical wavefronts, inverse-square law for intensity, and relation between amplitude and distance from a point source.

PGT extension

Rigorous derivation of the three-dimensional scalar wave equation in spherical polar coordinates and its divergent wave solutions of the form Psi(r,t) = (A/r)cos(kr - omega t).

Prerequisites

Simple harmonic motion and wave kinematics
Wave equation in one dimension
Vector calculus and spherical polar coordinate systems

Wavefronts and Rays in Three Dimensions

A wavefront is the locus of all adjacent points vibrating in the same phase, forming concentric spheres for a point source.
Wave normals (rays) point radially outward, indicating the direction of energy propagation.
As the radius r of the wavefront increases, the local surface area of the sphere grows as 4πr².

Amplitude Variation with Distance

Total wave power P crossing any spherical surface of radius r remains constant in a lossless medium.
Since intensity I is power per unit area, $I = P / (4\pir^{2})$, meaning intensity varies inversely as r².
Because wave intensity is proportional to the square of the amplitude (I ∝ A²), the displacement amplitude A must vary inversely with distance r: A(r) = A₀ / r.

The 3D Wave Equation in Spherical Coordinates

In regions devoid of sources and assuming spherical symmetry, the scalar wave equation depends only on the radial coordinate r and time t.
The Laplacian operator in spherical coordinates reduces to (1/r²) ∂/∂r(r² ∂/∂r).
The differential equation takes the form ∂²(rΨ)/∂t² = v² ∂²(rΨ)/∂r².

Energy Transport and Intensity

The average intensity of a spherical wave is given by I_avg = 2π² ρ v f² A₀² / r².
Spherical waves attenuate purely due to geometric spreading, independent of medium absorption (which causes exponential damping).
At distances very large compared to the source dimensions, spherical wavefronts can be approximated locally as planar wavefronts.

Derivations

Derivation of Spherical Wave Equation Solution

Conditions: Homogeneous, isotropic, lossless medium with spherical symmetry

Write the three-dimensional scalar wave equation: ∇²$\Psi = (1/v^{2})(\partial ^{2}\Psi/\partial t^{2})$.
Expand the Laplacian for spherically symmetric fields where $\Psi = \Psi(r$,t): (1/r²)∂/∂r(r² ∂Ψ/∂r) = (1/v²)(∂²Ψ/∂t²).
Substitute u(r,t) = rΨ(r,t). Then ∂Ψ/∂$r = (1/r)(\partial u/\partial r - u/r)$, leading to ∂²u/∂r² = (1/v²)(∂²u/∂t²).
Solve the resulting 1D wave equation for u: u(r,t) = f(r - vt) + g(r + vt).
Divide by r to obtain the spherical wave solution: Ψ(r,t) = (1/r)f(r - vt).

Result: \Psi(r,t) = \frac{A}{r} \cos(kr - \omega t)

Derivation of Inverse Square Law for Intensity

Conditions: Isotropic point source emitting constant average power P in a non-absorbing medium

Consider a point source emitting power P uniformly in all directions.
Enclose the source with an imaginary sphere of radius r. The total surface area of this sphere is $S = 4\pi r^2$.
Define intensity I as the power per unit area: $I = \frac{P}{S} = \frac{P}{4\pi r^2}$.
Express wave intensity in terms of displacement amplitude A: $I = \frac{1}{2} \rho v \omega^2 A^2$.
Equate the two expressions to show A^2 \propto \frac{1}{r^2}, yielding A \propto \frac{1}{r}.

Result: I(r) = \frac{P}{4\pi r^2}

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Spherical Wave Displacement Function	$$\Psi(r,t) = \frac{A_0}{r} \sin(kr - \omega t + \phi)$$	$A_0 = initial amplitude constant$, r = radial distance, k = propagation constant, \omega = angular frequency m (or field-specific units)	Isotropic, uniform medium, continuous sinusoidal point source
Wave Intensity from a Point Source	$$I = \frac{P}{4\pi r^2}$$	P = power of source, r = radial distance from source W/m^2	Lossless medium, omnidirectional point source
Amplitude Distance Relation	$$A(r) = \frac{A_1 r_1}{r}$$	$A_1 = amplitude at distance r_1$, r = target distance m	Conservation of wave energy in absence of absorption
Three-dimensional Scalar Laplacian (Symmetric)	$$\nabla^2 \Psi = \frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial \Psi}{\partial r}\right)$$	\Psi = wave function, r = radial coordinate m^{-2}	Purely radial dependence (spherical symmetry)
Transformed Radial Wave Equation	$$\frac{\partial^2 u}{\partial t^2} = v^2 \frac{\partial^2 u}{\partial r^2}$$	u = r\Psi, v = wave speed, t = time, r = distance m/s^2 (acceleration dimensions for u term)	Substitution u = r\Psi applied to spherical wave equation
Propagation Constant (Wave Number)	$$k = \frac{2\pi}{\lambda} = \frac{\omega}{v}$$	\lambda = wavelength, \omega = angular frequency, v = wave speed rad/m	Linear homogeneous medium
Average Energy Density	$$\langle E \rangle = \frac{1}{2} \rho \omega^2 A(r)^2$$	\rho = density of medium, \omega = angular frequency, A(r) = amplitude at r J/m^3	Harmonic spherical wave
Phase Velocity of Spherical Wave	$$v_p = \frac{\omega}{k} = f\lambda$$	$v_p = phase velocity$, f = frequency, \lambda = wavelength m/s	Non-dispersive medium

Graphs and experiments

Wavefronts and Rays

Draw: Concentric circles drawn around a central point source with orthogonal radial arrows pointing outwards labeled as rays.

Interpret: Demonstrates that wavefronts are spherical surfaces expanding outward, while rays indicate the direction of energy transport perpendicular to wavefronts.

Exam use: Tested in conceptual multiple-choice questions regarding the geometrical relationship between wavefronts and rays.

Amplitude vs Distance (A vs r)

Draw: A rectangular hyperbola curve plotting displacement amplitude A on the vertical axis against radial distance r on the horizontal axis.

Interpret: Illustrates the inverse relationship A ∝ 1/r, showing rapid drop-off near the source and gradual leveling further away.

Exam use: Used to interpret numerical scaling problems where distance is doubled or tripled.

Intensity vs Inverse Square Distance (I vs 1/r^2)

Draw: A straight line passing through the origin with a positive slope representing intensity I plotted against 1/r^2.

Interpret: Confirms the inverse-square law of wave intensity for a point source in a 3D medium.

Exam use: Frequently used in graphical analysis questions in PGT examinations.

Solved numericals

TGT core

If the amplitude of a spherical wave is 5.0 cm at a distance of 2.0 m from a point source, what is the amplitude at a distance of 10.0 m?

Identify the inverse proportionality between amplitude and distance for a spherical wave: A_1 $r_1 = A_2 r_2$.
Substitute the given values: $A_1 = 5.0 cm$, $r_1 = 2.0 m$, $r_2 = 10.0 m$.
Solve for A_2: $A_2 = (A_1 r_1) / r_2 = (5.0 cm * 2.0 m) / 10.0 m$.
Calculate the final value: $A_2 = 10 / 10 = 1.0 cm$.

Answer: 1.0 cm

PGT extension

A small acoustic point source radiates sound power uniformly at 31.4 W in an open field. Calculate the sound intensity and the displacement amplitude of air particles at a distance of 5.0 m. Take density of air $\rho = 1.2 kg/m^3$, speed of sound $v = 340 m/s$, and frequency f = 500 Hz.

Calculate intensity using $I = P / (4\pir^2): I = 31.4 / (4 * 3.1416 * 5^2) = 31.4 / 314.16 = 0.1 W/m^2$.
Relate intensity to amplitude using $I = (1/2) \rho v \omega^2 A^2$, where $\omega = 2\pif = 2 * 3.1416 * 500 = 3141.6 rad/s$.
Rearrange for amplitude A: $A = \sqrt(2I / (\rho v \omega^2))$.
Substitute values: $A = \sqrt((2 * 0.1) / (1.2 * 340 * (3141.6)^2)) = \sqrt(0.2 / (408 * 9869604)) = \sqrt(0.2 / 4.026 * 10^9) \approx 7.05 * 10^-6 m$.

Answer: Intensit$y = 0.1 W/m^2$, Amplitude ≈ 7.05 μm

PGT extension

At what distance from an isotropic point source does the intensity of a spherical wave drop to 1/9th of its value at 3.0 m?

Recall the inverse square law: I ∝ 1/r^2, which implies I_1 / $I_2 = (r_2 / r_1)^2$.
Given $I_2 = (1/9) I_1$, substitute this ratio into the equation: $9 = (r_2 / 3.0)^2$.
Take the square root on both sides: r_2 / 3.0 = 3.
Solve for r_2: $r_2 = 3 * 3.0 = 9.0 m$.

Answer: 9.0 m

Assumptions and limitations

Assumes the medium is homogeneous, isotropic, and infinite, neglecting boundary reflections and refraction.
Assumes a point source of zero physical dimensions; real sources have finite sizes where near-field deviations occur.
Does not account for attenuation due to medium absorption or scattering, which introduces an exponential damping factor e^{-\alpha r}.

Common misconceptions

Common error: Spherical wave intensity is inversely proportional to distance r.

Correct: Intensity is inversely proportional to the square of the distance (r^2).

Energy spreads over the surface area of a sphere, which grows as 4πr^2, not its radius.

Common error: Frequency of a spherical wave changes as it propagates outward.

Correct: Frequency remains constant and depends solely on the source.

Wave propagation through a stationary, linear medium alters amplitude and phase, but not frequency.

Common error: Spherical waves can only exist in three dimensions for sound.

Correct: Spherical waves occur in any three-dimensional isotropic medium, including light, sound, and seismic waves.

The geometric requirement of 1/r amplitude dependence arises purely from 3D divergence of wavefronts.

Quick revision

Spherical wavefronts emerge from point sources and expand with surface area 4πr^2.
Displacement amplitude decays as A(r) = A_0 / r.
Wave intensity obeys the inverse square law $I = P / (4\pir^2)$.
The 3D scalar wave equation reduces to a 1D wave equation via the substitution $u = r\Psi$.

Must remember

The displacement amplitude of a spherical wave varies inversely with distance $r$ from the source, given by $A(r) = \frac{A_0}{r}$.
The intensity of a spherical wave follows an inverse-square law with distance: $I = \frac{P}{4\pi r^2}$.
In spherical coordinates under purely radial symmetry, the Laplacian operator reduces to $\nabla^2 \Psi = \frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial \Psi}{\partial r}\right)$.
By substituting $u = r\Psi$, the spherical scalar wave equation transforms into a standard 1D wave equation: $\frac{\partial^2 u}{\partial t^2} = v^2 \frac{\partial^2 u}{\partial r^2}$.
Spherical waves attenuate purely due to geometric spreading in a lossless medium, independent of absorption damping.

Glossary

Spherical WavefrontThe locus of all adjacent points vibrating in the same phase, forming concentric spheres originating from a point source.

Wave Normal (Ray)A line drawn perpendicular to the wavefront pointing radially outward, indicating the direction of energy propagation.

Propagation Constant ($k$)The spatial frequency of the wave defined as $k = \frac{2\pi}{\lambda} = \frac{\omega}{v}$.

Geometric SpreadingThe reduction of wave intensity with distance due to the uniform expansion of wavefront surface area.

Plane Waves vs. Spherical Waves

Parameter	Plane Waves	Spherical Waves
Wavefront Geometry	Parallel planes	Concentric spheres
Amplitude Variation	Constant ($A_0$)	Inversely proportional to distance ($A_0/r$)
Intensity Variation	Constant with distance	Inversely proportional to square of distance ($1/r^2$)

Practice Quiz

Choose an option to reveal the explanation.

1. What is the geometric shape of wavefronts produced by a point source of light in a uniform isotropic medium?

Explanation: A point source radiates disturbances equally in all directions, creating concentric spheres of constant phase known as spherical wavefronts.

2. How does the displacement amplitude $A$ of a spherical wave change with distance $r$ from the source in a lossless medium?

Explanation: Since total power remains constant and intensity $I \propto A^2 \propto \frac{1}{r^2}$, the displacement amplitude must vary as $A(r) = \frac{A_0}{r}$.

3. If the distance from a point source emitting spherical waves is doubled, the intensity of the wave becomes:

Explanation: Intensity varies inversely as the square of the distance ($I \propto \frac{1}{r^2}$). When $r$ is doubled, intensity becomes $\frac{1}{(2r)^2} = \frac{1}{4}I$.

4. What is the form of the scalar Laplacian operator $\nabla^2 \Psi$ under purely radial symmetry in spherical coordinates?

Explanation: In regions of spherical symmetry where $\Psi$ depends only on $r$, the Laplacian reduces to $\nabla^2 \Psi = \frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial \Psi}{\partial r}\right)$.

5. By substituting $u = r\Psi$, the spherical wave equation transforms into which standard differential equation?

Explanation: Making the substitution $u = r\Psi$ transforms the radial part of the 3D wave equation into the standard 1D linear wave equation for $u(r,t)$.

6. What is the primary cause of wave attenuation for spherical waves propagating through a lossless medium?

Explanation: In a lossless medium, spherical waves attenuate solely due to geometric spreading, where constant total power is distributed over an ever-increasing surface area $4\pi r^2$.

7. The propagation constant $k$ of a wave is related to angular frequency $\omega$ and wave speed $v$ by:

Explanation: By definition, the wave number or propagation constant is $k = \frac{2\pi}{\lambda} = \frac{2\pi f}{v} = \frac{\omega}{v}$.

8. At distances very large compared to the dimensions of the source, spherical wavefronts can be approximated locally as:

Explanation: Far from the source, the radius of curvature of a spherical wavefront is extremely large, making a small local patch effectively indistinguishable from a plane wave.

9. What is the SI unit of the wave propagation constant $k$?

Explanation: The propagation constant $k = \frac{2\pi}{\lambda}$ has units of radians per meter (rad/m).

10. If the amplitude of a spherical wave at distance $r_1$ is $A_1$, what is its amplitude at distance $r_2$ in a lossless medium?

Explanation: Since amplitude varies inversely with distance ($A(r) = \frac{A_1 r_1}{r}$), the amplitude at $r_2$ is $A_1 \frac{r_1}{r_2}$.

11. Which of the following parameters remains constant as a spherical wave propagates outward through a stationary, uniform, lossless medium?

Explanation: Frequency depends exclusively on the source and remains constant during propagation, whereas amplitude, intensity, and energy density decrease with distance $r$.

12. What does the wave normal (ray) indicate in a spherical wave system?

Explanation: Wave normals point radially outward, representing the direction along which wave energy and wavefronts advance.

13. The average energy density $\langle E \rangle$ of a harmonic spherical wave is proportional to:

Explanation: Average energy density is given by $\langle E \rangle = \frac{1}{2} \rho \omega^2 A(r)^2$, showing it is directly proportional to the square of the displacement amplitude.

14. Which differential relation correctly defines the transformed variable $u = r\Psi$ substitution for spherical waves?

Explanation: Substitution of $u = r\Psi$ simplifies the 3D scalar wave equation into the standard 1D wave equation: $\frac{\partial^2 u}{\partial t^2} = v^2 \frac{\partial^2 u}{\partial r^2}$.

15. Which of the following media can support spherical waves?

Explanation: Spherical waves occur in any 3D isotropic medium whenever disturbances originate from a localized point source (e.g., light, sound, and seismic waves).

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. The intensity $I$ of a spherical wave at a radial distance $r$ from a point source of power $P$ in a lossless medium is given by:

Explanation: Intensity is power per unit area. For a spherical wavefront of radius $r$, the surface area is $4\pi r^2$, giving $I = \frac{P}{4\pi r^2}$.

2. If the displacement amplitude of a spherical wave at distance $r$ is $A(r) = \frac{A_0}{r}$, how does the wave intensity $I(r)$ vary with distance?

Explanation: Since wave intensity is proportional to the square of the displacement amplitude ($I \propto A^2$), substituting $A(r) = \frac{A_0}{r}$ yields $I(r) \propto \frac{1}{r^2}$.

3. In the analysis of three-dimensional scalar wave equation with spherical symmetry, the function $u(r,t) = r\Psi(r,t)$ satisfies which equation of motion?

Explanation: Applying the transformation $u = r\Psi$ to the spherically symmetric wave equation directly yields the standard one-dimensional wave equation for $u$: $\frac{\partial^2 u}{\partial t^2} = v^2 \frac{\partial^2 u}{\partial r^2}$.

4. A point source radiates sound power uniformly in all directions in a non-absorbing medium. If the amplitude of the sound wave at a distance of $2\text{ m}$ is $A$, what is the amplitude at a distance of $6\text{ m}$?

Explanation: Amplitude varies inversely with distance ($A \propto \frac{1}{r}$). When distance increases from $2\text{ m}$ to $6\text{ m}$ (a factor of 3), the amplitude decreases by a factor of 3, becoming $A / 3$.

5. Which of the following represents a valid divergent spherical wave solution for the displacement function $\Psi(r,t)$?

Explanation: A divergent spherical wave has an amplitude inversely proportional to $r$, represented by $\Psi(r,t) = \frac{A_0}{r} \sin(kr - \omega t)$.

6. What happens to the frequency of a spherical wave when it passes from one lossless isotropic medium into another with a different wave speed?

Explanation: Frequency is determined entirely by the source and remains invariant when a wave crosses boundaries into different media, whereas wavelength and speed change accordingly.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. The local surface area of a spherical wavefront of radius $r$ expands as:

Explanation: The surface area of a sphere of radius $r$ is given by $4\pi r^2$.

2. If the intensity of a spherical wave at distance $r_1$ is $I_1$, what is the intensity at distance $r_2 = 3r_1$?

Explanation: Intensity follows the inverse-square law: $I \propto \frac{1}{r^2}$. Thus, at $r_2 = 3r_1$, intensity is $I_1 / (3)^2 = I_1 / 9$.

3. In the spherical wave displacement equation $\Psi(r,t) = \frac{A_0}{r} \sin(kr - \omega t + \phi)$, what represents the spatial oscillation frequency?

Explanation: The propagation constant $k$ (wave number) determines the spatial oscillation frequency and phase progression along $r$.

4. The average energy density $\langle E \rangle$ of a spherical harmonic wave at distance $r$ varies with distance as:

Explanation: Energy density is proportional to $A(r)^2$. Since $A(r) \propto \frac{1}{r}$, the energy density varies as $\frac{1}{r^2}$.

5. Which coordinate system is naturally suited for formulating wave equations with central or point symmetry?

Explanation: Spherical polar coordinates exploit radial symmetry, simplifying the Laplacian operator when wave properties depend only on distance $r$.

6. What is the phase velocity $v_p$ of a spherical wave in terms of angular frequency $\omega$ and propagation constant $k$?

Explanation: Phase velocity is defined as $v_p = \frac{\omega}{k} = f\lambda$.

7. Why does a spherical wave decrease in amplitude as it propagates outward?

Explanation: Total wave power remains constant while spreading across a growing surface area ($4\pi r^2$), requiring amplitude to drop as $1/r$.

8. If a source of spherical waves has its power output doubled, what happens to the displacement amplitude $A(r)$ at any given distance $r$?

Explanation: Power $P$ is proportional to intensity $I$, and intensity is proportional to amplitude squared ($I \propto A^2$). If power doubles, $A^2$ doubles, so amplitude increases by a factor of $\sqrt{2}$.

9. Under what condition can a spherical wavefront be treated as a plane wavefront?

Explanation: When observed at a large distance from the source, the curvature of the spherical wavefront is negligible across a local region, allowing it to be approximated as a plane wave.

10. Which of the following is a false statement regarding spherical waves?

Explanation: Frequency does not vary with distance; it remains constant and is determined entirely by the source. Thus, option 3 is false.