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Waves Formulas for JEE 2026
Waves is one of the most important chapters in JEE Physics, with 2–3 questions asked every year. It includes key concepts like transverse and longitudinal waves, wave equation, standing waves, harmonics in strings and air columns, beats, Doppler effect, and speed of sound.
For effective preparation, using a JEE Mains Physics Formula PDF can help you quickly revise all important formulas and understand their applications, which is essential for scoring well in both JEE Mains and Advanced.
What Are Waves?
Drop a stone into a still pond and you see ripples spreading outward. The water itself doesn't travel to the edge — instead, a disturbance passes through the water. This disturbance, which carries energy from one place to another without transporting matter, is called a wave.
Wave: A disturbance that transfers energy through a medium (or through space) without the permanent displacement of the particles of the medium
Waves are everywhere in physics: sound, light, water ripples, vibrations on a string, seismic waves in the Earth, and even quantum matter waves. In this chapter, we focus on mechanical waves — waves that need a material medium to travel through.
Types of Waves: Transverse and Longitudinal
Waves are classified by the direction in which the particles of the medium move relative to the direction the wave travels.
Transverse Wave: A wave in which the particles of the medium vibrate perpendicular to the direction of wave propagation. Examples: waves on a string, light (electromagnetic waves).
Longitudinal Wave: A wave in which the particles of the medium vibrate parallel (back and forth) to the direction of wave propagation. Examples: sound waves, compression waves in a spring.
Comparison of Wave Types
| Property | Transverse | Longitudinal |
|---|---|---|
| Particle motion | Perpendicular to wave | Parallel to wave |
| Medium | Solids, surface of liquids | Solids, liquids, gases |
| Features | Crests and troughs | Compressions and rarefactions |
| Can be polarised? | Yes | No |
| Example | String vibration | Sound in air |
Wave Parameters
Every wave is described by a set of standard quantities.
Amplitude ($$A$$): The maximum displacement of a particle from its equilibrium position
Wavelength ($$\lambda$$): The distance between two consecutive points in the same phase (e.g., crest to crest or compression to compression). Measured in metres.
Frequency ($$f$$): The number of complete waves passing a point per second. Measured in hertz (Hz).
Wave Speed ($$v$$): The speed at which the wave disturbance travels through the medium
Fundamental Wave Relation
$$$v = f\lambda = \frac{\lambda}{T}$$$
where $$v$$ = wave speed, $$f$$ = frequency, $$\lambda$$ = wavelength, $$T = 1/f$$ = time period.
This relation holds for all types of waves.
The Wave Equation
A sinusoidal wave travelling along a string can be described mathematically by a function that gives the displacement of every point on the string at every instant of time.
Progressive Wave Equation
A sinusoidal wave travelling in the positive $$x$$-direction:
$$$y(x, t) = A\sin(kx - \omega t + \phi)$$$
A wave travelling in the negative $$x$$-direction:
$$$y(x, t) = A\sin(kx + \omega t + \phi)$$$
where:
- $$A$$ = amplitude
- $$k = 2\pi/\lambda$$ = wave number (in rad/m)
- $$\omega = 2\pi f = 2\pi/T$$ = angular frequency (in rad/s)
- $$\phi$$ = initial phase
- Wave speed: $$v = \omega/k = f\lambda$$
Note: The sign between $$kx$$ and $$\omega t$$ determines the direction: a "$$-$$" sign means the wave travels in the $$+x$$ direction; a "$$+$$" sign means it travels in the $$-x$$ direction. This is a common source of errors.
Worked Example
A wave is described by $$y = 0.02\sin(3x - 60t)$$ (in SI units). Find the amplitude, wavelength, frequency, and wave speed.
Comparing with $$y = A\sin(kx - \omega t)$$:
Amplitude: $$A = 0.02 \text{ m} = 2 \text{ cm}$$
Wave number: $$k = 3 \text{ rad/m} \Rightarrow \lambda = \frac{2\pi}{k} = \frac{2\pi}{3} = 2.09 \text{ m}$$
Angular frequency: $$\omega = 60 \text{ rad/s} \Rightarrow f = \frac{\omega}{2\pi} = \frac{60}{2\pi} = 9.55 \text{ Hz}$$
Wave speed: $$v = \frac{\omega}{k} = \frac{60}{3} = 20 \text{ m/s}$$ (travelling in $$+x$$ direction)
Wave Speed on a String
How fast a wave travels on a string depends on two things: how tight the string is (tension) and how heavy it is (mass per unit length). A tighter, lighter string carries waves faster.
Speed of a Transverse Wave on a String
$$$v = \sqrt{\frac{T}{\mu}}$$$
where $$T$$ = tension in the string (in N), $$\mu$$ = linear mass density = mass per unit length (in kg/m).
Worked Example
A string of length 2 m and mass 10 g is stretched with a tension of 200 N. Find the speed of transverse waves on this string.
$$\mu = \frac{m}{L} = \frac{0.01}{2} = 0.005 \text{ kg/m}$$
$$v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{200}{0.005}} = \sqrt{40000} = 200 \text{ m/s}$$
Superposition Principle for Waves
When two or more waves pass through the same region simultaneously, the net displacement at any point is simply the sum of the displacements due to each individual wave. This is the principle of superposition.
Principle of Superposition
$$$y_{\text{net}}(x, t) = y_1(x, t) + y_2(x, t) + \cdots$$$
The waves pass through each other unchanged after the overlap.
Superposition leads to two important phenomena: standing waves (when waves of the same frequency travel in opposite directions) and beats (when waves of slightly different frequencies overlap).
Standing Waves Formulas
When two identical waves travel in opposite directions along the same medium, they interfere to produce a pattern that appears to vibrate in place rather than travelling. This is a standing wave. Certain points never move (nodes), while other points vibrate with maximum amplitude (antinodes).
Standing Wave Equation
If $$y_1 = A\sin(kx - \omega t)$$ and $$y_2 = A\sin(kx + \omega t)$$, then:
$$$y = y_1 + y_2 = 2A\sin(kx)\cos(\omega t)$$$
- The amplitude at position $$x$$ is $$2A|\sin(kx)|$$ — it varies with position
- Nodes (zero amplitude): $$\sin(kx) = 0 \Rightarrow x = 0, \frac{\lambda}{2}, \lambda, \frac{3\lambda}{2}, \ldots$$
- Antinodes (maximum amplitude $$2A$$): $$|\sin(kx)| = 1 \Rightarrow x = \frac{\lambda}{4}, \frac{3\lambda}{4}, \frac{5\lambda}{4}, \ldots$$
- Distance between consecutive nodes = distance between consecutive antinodes = $$\lambda/2$$
- Distance between a node and the nearest antinode = $$\lambda/4$$
Node: A point on a standing wave that remains stationary (zero displacement at all times)
Antinode: A point on a standing wave where the displacement is maximum
Harmonics of Strings: Frequency Formulas
When a string is fixed at both ends (like a guitar string) and plucked, standing waves are formed. Both ends must be nodes (since they are fixed). Only certain wavelengths "fit" on the string — these are the allowed modes of vibration, called harmonics.
Harmonics of a String Fixed at Both Ends
For a string of length $$L$$:
$$$f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}} = \frac{nv}{2L}, \quad n = 1, 2, 3, \ldots$$$
- $$n = 1$$: Fundamental (1st harmonic), $$f_1 = \frac{v}{2L}$$, $$\lambda_1 = 2L$$
- $$n = 2$$: 2nd harmonic (1st overtone), $$f_2 = 2f_1$$, $$\lambda_2 = L$$
- $$n = 3$$: 3rd harmonic (2nd overtone), $$f_3 = 3f_1$$, $$\lambda_3 = 2L/3$$
- General: $$n$$th harmonic has $$n$$ loops (antinodes) and $$(n+1)$$ nodes
- All harmonics (odd and even) are present
Worked Example
A guitar string of length 60 cm is under 100 N tension and has linear mass density $$0.01$$ kg/m. Find the fundamental frequency and the first two overtones.
$$v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{100}{0.01}} = 100 \text{ m/s}$$
$$f_1 = \frac{v}{2L} = \frac{100}{2 \times 0.6} = 83.3 \text{ Hz}$$ (fundamental)
$$f_2 = 2f_1 = 166.7 \text{ Hz}$$ (1st overtone)
$$f_3 = 3f_1 = 250 \text{ Hz}$$ (2nd overtone)
Harmonics of Air Columns: Open and Closed Pipes
Sound waves can form standing waves inside tubes (organ pipes, flutes). The boundary conditions depend on whether the ends are open (pressure node, displacement antinode) or closed (pressure antinode, displacement node).
Open Pipe (Both Ends Open)
Open Pipe: All Harmonics Present
$$$f_n = \frac{nv}{2L}, \quad n = 1, 2, 3, \ldots$$$
- Fundamental: $$f_1 = \frac{v}{2L}$$, $$\lambda_1 = 2L$$
- Both odd and even harmonics are present
- Both ends are antinodes
Closed Pipe (One End Closed)
Closed Pipe: Only Odd Harmonics Present
$$$f_n = \frac{nv}{4L}, \quad n = 1, 3, 5, \ldots \text{ (odd only)}$$$
- Fundamental: $$f_1 = \frac{v}{4L}$$, $$\lambda_1 = 4L$$
- Only odd harmonics ($$n = 1, 3, 5, \ldots$$) are present
- Closed end is a node, open end is an antinode
- The fundamental frequency of a closed pipe is half that of an open pipe of the same length
Worked Example
An organ pipe of length 85 cm is open at both ends. The speed of sound is 340 m/s. Find: (a) the fundamental frequency, (b) the 3rd harmonic.
(a) $$f_1 = \frac{v}{2L} = \frac{340}{2 \times 0.85} = \frac{340}{1.7} = 200 \text{ Hz}$$
(b) $$f_3 = 3f_1 = 600 \text{ Hz}$$
Worked Example
The same pipe is now closed at one end. Find: (a) the new fundamental frequency, (b) the first overtone.
(a) $$f_1 = \frac{v}{4L} = \frac{340}{4 \times 0.85} = \frac{340}{3.4} = 100 \text{ Hz}$$
(b) First overtone = 3rd harmonic (since only odd harmonics exist):
$$f_3 = 3f_1 = 300 \text{ Hz}$$
Comparison: Open vs. Closed Pipe
| Property | Open Pipe | Closed Pipe |
|---|---|---|
| Fundamental $$f_1$$ | $$v/(2L)$$ | $$v/(4L)$$ |
| Harmonics present | All ($$n = 1, 2, 3, \ldots$$) | Odd only ($$n = 1, 3, 5, \ldots$$) |
| End conditions | Antinode–Antinode | Node–Antinode |
| Sound quality | Richer (more harmonics) | Hollow (missing evens) |
Tip: A common JEE trap: the "first overtone" is the second mode, not the second harmonic. For a closed pipe, the first overtone is the 3rd harmonic ($$3f_1$$), not the 2nd harmonic.
Beats Formula
When two sound waves of slightly different frequencies reach your ear simultaneously, you hear a sound that alternately grows louder and softer in a regular pattern. These periodic variations in loudness are called beats.
Beats: The periodic variation in the intensity of sound caused by the superposition of two waves of slightly different frequencies
Beat Frequency
If two waves have frequencies $$f_1$$ and $$f_2$$ (with $$f_1 \approx f_2$$):
$$$f_{\text{beat}} = |f_1 - f_2|$$$
The resultant wave:
$$$y = 2A\cos\left(\frac{\omega_1 - \omega_2}{2}t\right)\sin\left(\frac{\omega_1 + \omega_2}{2}t\right)$$$
- The amplitude varies with frequency $$|f_1 - f_2|/2$$
- You hear one beat each time the amplitude reaches maximum, so the beat frequency is $$|f_1 - f_2|$$
- Beats are audible only when $$|f_1 - f_2|$$ is small (typically $$< 7$$ Hz)
Worked Example
Two tuning forks have frequencies 256 Hz and 260 Hz. How many beats per second are heard?
$$f_{\text{beat}} = |260 - 256| = 4 \text{ beats per second}$$
The listener hears 4 moments of loudness per second.
Worked Example
A tuning fork of frequency 512 Hz produces 5 beats per second with a string. When the tension in the string is increased, the beat frequency decreases to 3 beats/s. Find the original frequency of the string.
The string frequency is either $$512 + 5 = 517$$ Hz or $$512 - 5 = 507$$ Hz.
When tension increases, the frequency of the string increases (since $$f \propto \sqrt{T}$$).
If the original frequency were 517 Hz, increasing it would make beats increase (moving further from 512). But beats decrease, so the original frequency must be 507 Hz.
After tension increase: $$f = 512 - 3 = 509$$ Hz or $$512 + 3 = 515$$ Hz. Since frequency increased from 507, it is now 509 Hz. ✓
Doppler Effect Formula
You have probably noticed that an ambulance siren sounds higher-pitched as it approaches and lower-pitched as it moves away. This change in the observed frequency due to relative motion between the source and the observer is called the Doppler effect.
Doppler Effect: The apparent change in frequency (or wavelength) of a wave due to relative motion between the source of the wave and the observer
Doppler Effect Formula for Sound
$$$f' = f_0 \left(\frac{v \pm v_o}{v \mp v_s}\right)$$$
where:
- $$f'$$ = observed (apparent) frequency
- $$f_0$$ = actual frequency of the source
- $$v$$ = speed of sound in the medium
- $$v_o$$ = speed of the observer
- $$v_s$$ = speed of the source
Sign convention:
- Numerator: $$+v_o$$ if observer moves towards source; $$-v_o$$ if away
- Denominator: $$-v_s$$ if source moves towards observer; $$+v_s$$ if away
Memory aid: "Towards" gives a higher frequency. In the formula, "towards" makes the fraction larger (add in numerator, subtract in denominator).
Worked Example
A car moving at 30 m/s sounds a horn of frequency 500 Hz. Find the frequency heard by a stationary observer (a) in front of the car, (b) behind the car. Take $$v = 340$$ m/s.
(a) Source approaching observer:
$$f' = 500 \times \frac{340}{340 - 30} = 500 \times \frac{340}{310} = 548.4 \text{ Hz}$$
(b) Source receding from observer:
$$f' = 500 \times \frac{340}{340 + 30} = 500 \times \frac{340}{370} = 459.5 \text{ Hz}$$
Worked Example
Both the source (400 Hz) and observer move towards each other at 20 m/s each. Find the apparent frequency. ($$v = 340$$ m/s)
$$f' = 400 \times \frac{340 + 20}{340 - 20} = 400 \times \frac{360}{320} = 400 \times 1.125 = 450 \text{ Hz}$$
Tip: JEE shortcut: if only the source moves with speed $$v_s \ll v$$, then $$\Delta f/f \approx v_s/v$$. If the source approaches, frequency increases by roughly this fraction.
Characteristics of Sound: Intensity, Loudness, Pitch
When we hear sound, we perceive three distinct qualities: how "high" or "low" it sounds, how "loud" or "soft" it is, and what makes a piano sound different from a violin playing the same note. These correspond to three physical properties.
Characteristics of Sound
| Perception | Physical Property | Depends On |
|---|---|---|
| Pitch | Frequency | Higher frequency = higher pitch |
| Loudness | Intensity (amplitude) | Greater amplitude = louder |
| Quality (timbre) | Waveform (harmonics) | Combination of overtones |
Intensity of Sound
$$$I = \frac{P}{4\pi r^2}$$$
where $$P$$ = power of the source, $$r$$ = distance from the source.
- Intensity follows the inverse square law: $$I \propto 1/r^2$$
- Intensity level (in decibels): $$\beta = 10\log_{10}\left(\frac{I}{I_0}\right)$$ dB
- $$I_0 = 10^{-12} \text{ W/m}^2$$ (threshold of human hearing)
Speed of Sound Formulas
The speed at which sound travels depends on the properties of the medium. Sound travels fastest in solids, slower in liquids, and slowest in gases — because the molecules in solids are more tightly packed and can transmit vibrations more quickly.
Speed of Sound in Different Media
In a gas (Newton–Laplace formula):
$$$v = \sqrt{\frac{\gamma P}{\rho}} = \sqrt{\frac{\gamma RT}{M}}$$$
where $$\gamma = C_p/C_v$$, $$P$$ = pressure, $$\rho$$ = density, $$T$$ = temperature (K), $$M$$ = molar mass.
In a solid rod:
$$$v = \sqrt{\frac{Y}{\rho}}$$$
where $$Y$$ = Young's modulus.
In a liquid:
$$$v = \sqrt{\frac{B}{\rho}}$$$
where $$B$$ = bulk modulus.
Factors affecting speed of sound in air:
- Temperature: $$v \propto \sqrt{T}$$. At $$0^\circ$$C, $$v \approx 332$$ m/s. At $$20^\circ$$C, $$v \approx 343$$ m/s.
- Approximate formula: $$v \approx (332 + 0.6\,t)$$ m/s, where $$t$$ is temperature in $$^\circ$$C.
- Humidity: Moist air is less dense than dry air (water vapour is lighter than $$\text{N}_2$$ and $$\text{O}_2$$), so sound is faster in humid air.
- Pressure: At constant temperature, changing pressure does not affect the speed of sound (because $$P/\rho$$ remains constant for an ideal gas).
Worked Example
Find the speed of sound in air at $$27^\circ$$C. Take $$\gamma = 1.4$$, $$M = 29 \times 10^{-3}$$ kg/mol.
$$v = \sqrt{\frac{\gamma RT}{M}} = \sqrt{\frac{1.4 \times 8.314 \times 300}{29 \times 10^{-3}}}$$
$$= \sqrt{\frac{3491.9}{0.029}} = \sqrt{120411} = 347 \text{ m/s}$$
Worked Example
The speed of sound at $$0^\circ$$C is 332 m/s. Find the speed at $$100^\circ$$C.
$$\frac{v_2}{v_1} = \sqrt{\frac{T_2}{T_1}} = \sqrt{\frac{373}{273}} = \sqrt{1.366} = 1.169$$
$$v_2 = 332 \times 1.169 = 388 \text{ m/s}$$
Tip: Speed of sound: Solids (~5000 m/s) > Liquids (~1500 m/s) > Gases (~340 m/s). In JEE, if a problem involves sound in air and doesn't specify the speed, use $$v = 340$$ m/s or $$v = 330$$ m/s as mentioned in the problem.
Key Formulas at a Glance
Quick Reference
- Wave relation: $$v = f\lambda$$
- Progressive wave: $$y = A\sin(kx - \omega t)$$, $$v = \omega/k$$
- Speed on string: $$v = \sqrt{T/\mu}$$
- Standing wave: $$y = 2A\sin(kx)\cos(\omega t)$$
- String (both ends fixed): $$f_n = nv/(2L)$$, all $$n$$
- Open pipe: $$f_n = nv/(2L)$$, all $$n$$
- Closed pipe: $$f_n = nv/(4L)$$, odd $$n$$ only
- Beats: $$f_{\text{beat}} = |f_1 - f_2|$$
- Doppler: $$f' = f_0\left(\frac{v \pm v_o}{v \mp v_s}\right)$$
- Speed of sound in gas: $$v = \sqrt{\gamma RT/M}$$
Waves Formulas for JEE 2026: Conclusion
Waves formulas for JEE 2026 are very important for quick revision and strong exam preparation. As discussed, this chapter includes key topics like wave equation, standing waves, harmonics in strings and air columns, beats, Doppler effect, and speed of sound. Since the chapter has both concept-based and formula-based questions, regular revision of the main formulas can help students solve problems faster and with better accuracy in the exam.
In the final phase of preparation, students should focus on understanding the meaning of each formula instead of only memorising it. Special attention should be given to common high-weightage areas like harmonics, beats, and Doppler effect. With a clear formula sheet, short notes, and regular practice, Waves can become one of the most scoring chapters in JEE Physics 2026.
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