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Waves and Oscillations

Resonance in Oscillatory Systems

Resonance is a physical phenomenon in which a vibrating system or external force drives another system to oscillate with greater amplitude at specific frequencies. It occurs when the frequency of an externally applied periodic force matches the natural frequency of the driven system.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Definition of resonance, condition for resonance, mechanical and acoustic resonance, examples such as tuning forks, simple pendulum, and basic resonance tube experiments.

PGT extension

Mathematical treatment of forced oscillations, amplitude and phase resonance, sharpness of resonance, quality factor (Q-factor), electrical LCR resonance circuits, and power absorption.

Prerequisites

Simple Harmonic Motion (SHM)
Damped harmonic oscillations and damping coefficient
Forced oscillations and driving frequency

Condition for Resonance

Resonance occurs in a forced oscillator when the angular frequency of the driving force (omega) equals the natural frequency (omega_0) of the system.
At resonance, the velocity of the oscillating system is exactly in phase with the driving force, resulting in maximum power transfer.
The amplitude of oscillation reaches its maximum value, limited solely by the damping present in the system.

Amplitude and Phase at Resonance

The steady-state amplitude of a forced damped oscillator is inversely proportional to the damping coefficient when omeg$a = \omega_0$.
The phase difference between the displacement and the driving force is exactly pi/2 radians (90 degrees) at resonance.
The maximum amplitude does not occur precisely at omeg$a = \omega_0 when damping$ is present, but at a slightly lower frequency given by omeg$a = \sqrt(\omega_0^2 - 2beta^2)$.

Sharpness of Resonance and Q-Factor

The sharpness of the resonance curve describes how rapidly the amplitude drops as the driving frequency deviates from the natural frequency.
The Quality factor (Q-factor) is a dimensionless parameter that measures the sharpness of resonance and energy storage relative to energy dissipation.
A system with light damping exhibits a high Q-factor, a very sharp resonance peak, and sustained oscillations.

Acoustic and Electrical Resonance

Acoustic resonance occurs in air columns, such as in Kundt's tube or resonance column apparatus, producing stationary longitudinal waves with displacement nodes and anti-nodes.
Electrical resonance occurs in series or parallel LCR circuits when the inductive reactance equals the capacitive reactance ($X_L = X_C)$.
At electrical resonance, the impedance of a series LCR circuit is purely resistive and reaches its minimum value, maximizing the circuit current.

Derivations

Amplitude and Phase of Forced Damped Oscillations

Conditions: A particle of mass m subjected to restoring force -kx, damping force -bv, and driving force F_0 cos(omega t).

Write the equation of motion: m(d^2x/dt^2) + b(dx/dt) + k$x = F_0 \cos(\omega t)$.
Substitute trial solution $x = A \cos(\omega t - \phi)$ and solve for steady-state amplitude $A = (F_0/m) / \sqrt((\omega_0^2 - \omega^2)^2 + (4 \beta^2 \omega^2))$, where omeg$a_0^2 = k/m$ and bet$a = b/(2m)$.
Determine the phase lag phi between displacement and driving force from tan(phi) = (2 beta omega) / (omega_0^2 - omega^2).

Result: $A = (F_0/m) / \sqrt((\omega_0^2 - \omega^2)^2 + 4 \beta^2 \omega^2)$ and tan(phi) = 2 beta omega / (omega_0^2 - omega^2)

Resonant Frequency and Quality Factor of Series LCR Circuit

Conditions: Alternating voltage source connected to a resistor R, inductor L, and capacitor C in series.

Write the total impedance Z of the circuit: $Z = R + j(\omega L - 1/(\omega C))$.
Apply the resonance condition where the imaginary part of impedance is zero: omega_r $L = 1/(\omega_r C)$.
Solve for resonant angular frequency omeg$a_r = 1/\sqrt(LC)$ and define the quality factor $Q = \omega_r L / R$.

Result: omeg$a_r = 1 / \sqrt{LC}$ and Q = \frac{1}{R} \sqrt{\frac{L}{C}}

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Natural Angular Frequency	$$\omega_0 = \sqrt{\frac{k}{m}}$$	k = spring constant, m = mass rad/s	Undamped free oscillation
Amplitude of Forced Oscillation	$$A = \frac{F_0 / m}{\sqrt{(\omega_0^2 - \omega^2)^2 + 4\beta^2\omega^2}}$$	$F_0 = driving force amplitude$, m = mass, beta = damping factor, omega = driving frequency, omeg$a_0 = natural frequency$ m	Steady-state forced oscillation
Phase Angle at Resonance	$$\tan(\phi) = \frac{2\beta\omega}{\omega_0^2 - \omega^2}$$	phi = phase difference between displacement and driving force radians or degrees	Forced damped system
Quality Factor (Q-factor)	$$Q = \frac{\omega_r}{2\beta} = \frac{\omega_r L}{R}$$	omeg$a_r = resonant frequency$, beta = damping coefficient, L = inductance, R = resistance Dimensionless	Light damping limit
Series LCR Resonant Frequency	$$f_r = \frac{1}{2\pi \sqrt{LC}}$$	L = inductance, C = capacitance Hz	$X_L = X_C$
Bandwidth of Resonance	$$\Delta \omega = \omega_2 - \omega_1 = \frac{R}{L}$$	omega_1 and omeg$a_2 = half-power frequencies$ rad/s	High Q approximation
Resonance Condition in Acoustic Column	$$l_1 + c = \frac{\lambda}{4}, \quad l_2 + c = \frac{3\lambda}{4}$$	l_1, $l_2 = first$ and second resonant lengths, c = end correction, lambda = wavelength m	Closed organ pipe / resonance tube
Power Absorbed at Resonance	$$P_{avg} = \frac{F_0^2}{2b}$$	$F_0 = driving force amplitude$, b = damping constant Watts (W)	At omeg$a = \omega_0$

Graphs and experiments

Amplitude-Frequency Resonance Curves

Draw: A family of curves plotting steady-state amplitude (A) versus driving frequency (omega) for different values of damping coefficient (b).

Interpret: Shows that as damping decreases, the resonance peak becomes higher and narrower. The maximum amplitude shifts slightly to the left of omega_0 for larger damping.

Exam use: Frequently asked in conceptual questions regarding the effect of damping on resonance height and sharpness.

Resonance Column Experiment

Draw: A vertical glass tube filled with water, connected to a reservoir, with a vibrating tuning fork held above the open mouth.

Interpret: Determines the velocity of sound in air by locating successive resonant lengths (l_1 and l_2) corresponding to stationary wave anti-nodes at the open end.

Exam use: Core practical experiment for TGT and PGT examinations, testing end correction and wavelength calculation.

Series LCR Resonance Curve (Current vs Frequency)

Draw: A bell-shaped curve showing current (I) on the y-axis against frequency (f) on the x-axis, featuring half-power points at I_max / sqrt(2).

Interpret: Illustrates frequency selectivity in electrical circuits. The distance between half-power frequencies defines the bandwidth.

Exam use: Tested in numericals involving bandwidth, resonant frequency, and quality factor calculation.

Solved numericals

TGT core

A resonance air column tube shows resonance with a tuning fork of frequency 512 Hz at length 16.5 cm and 52.5 cm. Calculate the velocity of sound in air.

Recall the relation between wavelength and successive resonant lengths: lambd$a = 2(l_2 - l_1)$.
Substitute the given lengths: lambd$a = 2(0.525 m - 0.165 m) = 2(0.360 m) = 0.720 m$.
Calculate the velocity of sound using $v = f * \lambda: v = 512 Hz * 0.720 m = 368.64 m/s$.

Answer: 368.64 m/s

PGT extension

A series LCR circuit has an inductance L = 2 H, capacitance C = 32 microF, and resistance R = 10 ohms. Find the resonant frequency and the quality factor Q of the circuit.

Calculate resonant angular frequency omeg$a_r = 1 / \sqrt(LC) = 1 / \sqrt(2 * 32 * 10^-6) = 1 / \sqrt(64 * 10^-6) = 1 / (8 * 10^-3) = 125 rad/s$.
Convert to linear frequency $f_r = \omega_r / (2 * \pi) = 125 / (2 * 3.1416) = 19.89 Hz$.
Calculate Q-factor using $Q = \omega_r L / R = (125 * 2) / 10 = 250 / 10 = 25$.

Answer: Resonant frequency = 19.89 Hz, Q-factor = 25

PGT extension

A forced oscillator has a mass of 1 kg, natural frequency 200 rad/s, and damping constant $b = 2 kg/s$. If a periodic driving force of peak value 10 N acts on it at resonance, calculate the maximum steady-state amplitude.

At resonance (omeg$a = \omega_0)$, the amplitude formula simplifies to $A_max = F_0 / (2 \beta \omega_0) or F_0 / (b \omega_0)$.
Identify values: $F_0 = 10 N$, $b = 2 kg/s$, omeg$a_0 = 200 rad/s$.
Substitute values into the formula: $A_max = 10 / (2 * 200) = 10 / 400 = 0.025 m$.

Answer: 0.025 m (or 2.5 cm)

Assumptions and limitations

Linearity assumption: Restoring forces are strictly proportional to displacement (Hooke's law holds), which fails at large amplitudes where anharmonic terms become significant.

Common misconceptions

Common error: Maximum amplitude always occurs exactly at the natural frequency omega_0.

Correct: Maximum amplitude occurs at omeg$a = \sqrt(\omega_0^2 - 2beta^2)$, which is slightly lower than omega_0 in damped systems.

Damping introduces a velocity-dependent term that shifts the peak frequency of the response curve.

Common error: At resonance, the phase difference between displacement and driving force is zero.

Correct: At resonance, the phase difference between displacement and driving force is exactly pi/2 (90 degrees).

The velocity is in phase with the driving force at resonance, meaning displacement lags velocity by pi/2 and thus lags the driving force by pi/2.

Quick revision

Resonance occurs when the driving frequency matches the natural frequency, maximizing amplitude and energy transfer.
Natural Angular Frequency: \omeg$a_0 = \sqrt{\frac{k}{m}}$
Amplitude of Forced Oscillation: $A = \frac{F_0 / m}{\sqrt{(\omega_0^2 - \omega^2)^2 + 4\beta^2\omega^2}}$

Must remember

Resonance occurs when the driving frequency $\omega$ equals the natural frequency $\omega_0$ of the oscillatory system.
At resonance, the velocity of the system is exactly in phase with the driving force, leading to maximum power transfer.
The phase difference between displacement and the driving force is exactly $\pi/2$ radians ($90^\circ$) at resonance.
With damping present, the maximum amplitude occurs at a frequency slightly lower than natural frequency: $\omega = \sqrt{\omega_0^2 - 2\beta^2}$.
The Quality factor ($Q$-factor) measures the sharpness of resonance and is defined as $Q = \frac{\omega_r}{2\beta} = \frac{\omega_r L}{R}$.
In series LCR circuits, electrical resonance occurs when inductive reactance equals capacitive reactance ($X_L = X_C$).
Bandwidth of resonance is given by $\Delta \omega = \omega_2 - \omega_1 = \frac{R}{L}$ for high $Q$ systems.
Acoustic resonance in a resonance tube produces stationary longitudinal waves with nodes and antinodes.

Glossary

Natural Frequency ($\omega_0$)The frequency at which a system oscillates freely without any external driving force or damping, given by $\sqrt{k/m}$.

Damping Factor ($\beta$)A parameter representing resistive forces that dissipate energy in an oscillatory system over time.

Quality Factor ($Q$)A dimensionless parameter indicating the sharpness of the resonance peak and the ratio of stored energy to dissipated energy.

Bandwidth ($\Delta \omega$)The frequency range between the two half-power frequencies where power drops to half of its maximum value.

End Correction ($c$)A small correction added to the actual length of an acoustic tube to account for the antinode forming slightly outside the open end.

Series LCR Resonance vs Parallel LCR Resonance

Parameter	Series LCR Resonance	Parallel LCR Resonance
Impedance at Resonance	Minimum ($Z = R$)	Maximum
Current at Resonance	Maximum	Minimum
Condition	$X_L = X_C$	$X_L = X_C$

Undamped vs Damped Forced Resonance Frequency

Feature	Undamped System	Damped System
Peak Amplitude Frequency	$\omega = \omega_0$	$\omega = \sqrt{\omega_0^2 - 2\beta^2}$
Phase at Resonance ($\omega = \omega_0$)	$\pi/2$	$\pi/2$
Max Amplitude Value	Infinite (if unconstrained)	Finite ($F_0 / (2m\beta\omega_0)$)

Practice Quiz

Choose an option to reveal the explanation.

1. What is the condition for resonance in a forced oscillator?

Explanation: Resonance occurs when the angular frequency of the driving force $\omega$ matches the natural frequency $\omega_0$ of the system.

2. What is the phase difference between displacement and the driving force at resonance?

Explanation: At resonance, velocity is in phase with the driving force, meaning displacement lags the driving force by exactly $\pi/2$ radians ($90^\circ$).

3. In a damped forced oscillator, the maximum amplitude occurs at which frequency?

Explanation: Due to damping, the peak amplitude is shifted slightly lower than the natural frequency to $\omega = \sqrt{\omega_0^2 - 2\beta^2}$.

4. What does a high Quality factor ($Q$-factor) indicate for an oscillatory system?

Explanation: A high $Q$-factor indicates light damping, slow energy dissipation, and a very sharp resonance peak.

5. What is the resonant frequency formula for a series LCR circuit?

Explanation: The resonant frequency of a series LCR circuit occurs when $X_L = X_C$, yielding $f_r = \frac{1}{2\pi \sqrt{LC}}$.

6. At electrical resonance in a series LCR circuit, the impedance of the circuit is:

Explanation: At resonance, $X_L = X_C$, canceling each other out, making the total impedance minimum and equal purely to resistance $R$.

7. What is the bandwidth ($\Delta \omega$) of a high-$Q$ resonance circuit with inductance $L$ and resistance $R$?

Explanation: The bandwidth between half-power frequencies for a high-$Q$ circuit is given by $\Delta \omega = \frac{R}{L}$.

8. Which of the following represents the expression for the average power absorbed at resonance?

Explanation: The average power absorbed at resonance in a forced oscillator is $P_{avg} = \frac{F_0^2}{2b}$, where $b$ is the damping constant.

9. Acoustic resonance in a resonance tube apparatus produces:

Explanation: Acoustic resonance in air columns establishes stationary longitudinal waves with well-defined displacement nodes and antinodes.

10. If the spring constant of a mechanical oscillator is $k$ and mass is $m$, what is its natural angular frequency?

Explanation: The natural angular frequency for a mass-spring system is given by $\omega_0 = \sqrt{\frac{k}{m}}$.

11. In the resonance tube experiment, the first and second resonant lengths are $l_1$ and $l_2$ including end correction $c$. The wavelength $\lambda$ is equal to:

Explanation: Since $l_1 + c = \lambda/4$ and $l_2 + c = 3\lambda/4$, subtracting the two gives $l_2 - l_1 = \lambda/2$, hence $\lambda = 2(l_2 - l_1)$.

12. The $Q$-factor of a series LCR circuit containing resistance $R$, inductance $L$, and capacitance $C$ can be written as:

Explanation: The quality factor $Q$ is defined as the ratio of inductive reactance to resistance at resonance: $Q = \frac{\omega_r L}{R}$.

13. What happens to the current in a series LCR circuit when it is tuned to resonance?

Explanation: Since impedance is minimum at resonance ($Z = R$), the current $I = V/R$ reaches its maximum possible value.

14. Which factor exclusively limits the maximum amplitude of a forced oscillator at exact resonance in an idealized undamped system?

Explanation: In an undamped system at resonance ($\omega = \omega_0$), the denominator becomes zero, resulting in theoretically infinite amplitude unless constrained by damping.

15. The dimension of the $Q$-factor is:

Explanation: The $Q$-factor is a ratio of energies or reactances to resistance, making it entirely dimensionless.

PYQ Patterns

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

1. In a damped forced harmonic oscillator, the phase angle $\phi$ between displacement and the driving force at resonance frequency is:

Explanation: At resonance ($\omega = \omega_0$), the term $\omega_0^2 - \omega^2$ becomes zero, making $\tan(\phi) = \infty$, which corresponds to a phase angle of $\pi/2$ radians.

2. A series LCR circuit has $L = 2\text{ H}$, $C = 32\text{ \mu F}$, and $R = 10\text{ }\Omega$. What is the resonant frequency of the circuit?

Explanation: Using $f_r = \frac{1}{2\pi \sqrt{LC}} = \frac{1}{2\pi \sqrt{2 \times 32 \times 10^{-6}}} = \frac{1}{2\pi \times 8 \times 10^{-3}} = \frac{125}{16\pi} \approx 62.5\text{ Hz}$ (approx calculation taking $\pi^2 \approx 10$ or standard values).

3. The sharpness of resonance in a series LCR circuit is increased by:

Explanation: Since $Q = \frac{\omega_r L}{R}$, decreasing resistance $R$ increases the $Q$-factor, leading to a sharper resonance curve.

4. In a resonance column experiment, the first resonant length is $17\text{ cm}$ and the second resonant length is $52\text{ cm}$. The end correction is:

Explanation: We know $l_2 + c = 3(l_1 + c)$. Substituting $52 + c = 3(17 + c) \Rightarrow 52 + c = 51 + 3c \Rightarrow 2c = 1 \Rightarrow c = 0.5\text{ cm}$? Wait, let's recalculate: $l_2 - l_1 = \lambda/2 = 52 - 17 = 35\text{ cm}$, so $\lambda = 70\text{ cm}$. Then $l_1 + c = \lambda/4 = 70/4 = 17.5\text{ cm}$. Since $l_1 = 17\text{ cm}$, end correction $c = 17.5 - 17 = 0.5\text{ cm}$.

5. At resonance, the power factor of a series LCR circuit is:

Explanation: At resonance, reactance is zero, making impedance entirely resistive ($Z = R$). Thus, phase angle is zero, and power factor $\cos(\phi) = 1$.

6. The maximum amplitude of a forced damped oscillator at resonance is inversely proportional to:

Explanation: The steady-state amplitude at resonance is $A_{max} = \frac{F_0}{2m\beta\omega_0}$, which is inversely proportional to the damping coefficient $\beta$.

Timed Topic Test

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

1. What is the primary effect of increasing damping on the resonance frequency and peak amplitude?

Explanation: Increased damping lowers the peak amplitude and shifts the maximum amplitude frequency to a lower value given by $\sqrt{\omega_0^2 - 2\beta^2}$.

2. In a parallel LCR circuit, resonance is characterized by:

Explanation: Unlike series LCR circuits, parallel LCR resonance maximizes impedance and minimizes total supply current at resonance.

3. If the driving frequency is much higher than the natural frequency ($\omega \gg \omega_0$) in a forced oscillator, the displacement is:

Explanation: At high driving frequencies ($\omega \gg \omega_0$), the inertial term dominates, causing the displacement to lag the driving force by $\pi$ radians.

4. What is the unit of the damping coefficient $\beta$ in forced oscillations?

Explanation: The damping term appears as $2\beta\dot{x}$, where $\beta$ has dimensions of frequency ($\text{s}^{-1}$).

5. The half-power frequencies $\omega_1$ and $\omega_2$ are defined as frequencies where the average power is what fraction of the maximum power at resonance?

Explanation: At half-power frequencies, the power absorbed by the system drops to half ($1/2$) of its maximum value at resonance.

6. Which of the following acoustic devices relies heavily on resonance to amplify specific frequencies of sound?

Explanation: Stethoscopes, Kundt's tubes, resonance columns, and Helmholtz resonators all operate on the principle of acoustic resonance.

7. If the quality factor $Q$ of a circuit is 50 and its resonant frequency is $500\text{ rad/s}$, what is the bandwidth?

Explanation: Since $Q = \frac{\omega_r}{\Delta \omega}$, the bandwidth $\Delta \omega = \frac{\omega_r}{Q} = \frac{500}{50} = 10\text{ rad/s}$.

8. Why can marching troops break a suspension bridge if they march in step?

Explanation: Matching steps at the bridge's natural frequency induces mechanical resonance, leading to dangerously large oscillation amplitudes.

9. In an undamped forced oscillation, what is the amplitude of the system when the driving frequency $\omega$ approaches the natural frequency $\omega_0$?

Explanation: Without damping to dissipate energy, the denominator in the amplitude formula approaches zero at $\omega = \omega_0$, causing amplitude to tend to infinity.

10. The velocity resonance (maximum velocity) in a forced damped oscillator occurs exactly when:

Explanation: Unlike displacement amplitude, velocity amplitude reaches its absolute maximum precisely at the natural frequency $\omega = \omega_0$, where velocity is in phase with the driving force.