Waves
Complete Formula Sheet & Shortcut Bible · BITSAT 2026
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Progressive Sinusoidal Waves
General Equation
y(x, t) = A sin(kx - ωt + φ₀)
+x dir: kx - ωt
-x dir: kx + ωt
+x dir: kx - ωt
-x dir: kx + ωt
Wave Parameters
Wave Number (k) = 2π/λ
Angular Freq (ω) = 2πf
Wave Velocity (v) = fλ = ω/k
Angular Freq (ω) = 2πf
Wave Velocity (v) = fλ = ω/k
Particle Velocity & Acceleration
vₚ = dy/dt = Aω cos(kx - ωt + φ₀)
aₚ = dvₚ/dt = -Aω² sin(kx - ωt + φ₀)
aₚ = dvₚ/dt = -Aω² sin(kx - ωt + φ₀)
Key Relations
Max Particle Velocity: (vₚ)ₘₐₓ = Aω
vₚ = -v × (slope of y-x graph)
vₚ = -v × (slope of y-x graph)
Wave Properties & Superposition
Velocity on String
v = √(T/μ)
T = Tension
μ = mass/length
T = Tension
μ = mass/length
Intensity & Energy
Intensity (I) ∝ A²f²
I = P / (4πr²) for point source
I = P / (4πr²) for point source
Phase & Path Difference
Phase Diff (Δφ) = 2π λ × Path Diff (Δx)
Resultant Amplitude
Aᵣ = √(A₁² + A₂² + 2A₁A₂cosφ)
Max (Constructive): A₁ + A₂
Min (Destructive): |A₁ - A₂|
Max (Constructive): A₁ + A₂
Min (Destructive): |A₁ - A₂|
Beats
Beat Frequency
fbeat = |f₁ - f₂|
fbeat = |f₁ - f₂|
Standing Waves
Equation
y(x, t) = (2A sin kx) cos ωt
Amplitude at x: A(x) = 2A sin kx
Amplitude at x: A(x) = 2A sin kx
Nodes (Zero Amp.)
Position: x = nλ/2
(n = 0, 1, 2, ...)
(n = 0, 1, 2, ...)
Antinodes (Max Amp.)
Position: x = (2n+1)λ/4
(n = 0, 1, 2, ...)
(n = 0, 1, 2, ...)
Key Distances
Node to Node = λ/2
Antinode to Antinode = λ/2
Node to Antinode = λ/4
Antinode to Antinode = λ/2
Node to Antinode = λ/4
Doppler Effect
General Formula (Sound)
f' = f [ (v ± vₒ) / (v ∓ vₛ) ]
v: speed of sound
vₒ: observer speed
vₛ: source speed
v: speed of sound
vₒ: observer speed
vₛ: source speed
Sign Convention
Use UPPER sign for TOWARDS motion.
Use LOWER sign for AWAY motion.
Use LOWER sign for AWAY motion.
Doppler Effect (Light)
Apparent Freq: f' = f √( (1+β)/(1-β) )
where β = v/c
Approx: Δλ/λ ≈ vᵣ/c (for vᵣ << c)
where β = v/c
Approx: Δλ/λ ≈ vᵣ/c (for vᵣ << c)
Special Cases
Source at rest: f' = f(1 ± vₒ/v)
Observer at rest: f' = f / (1 ∓ vₛ/v)
Observer at rest: f' = f / (1 ∓ vₛ/v)
BITSAT Wave Tricks
Wave Speed from Equation
For y = f(ax ± bt),
Wave Speed v = |b/a|
(coeff of t / coeff of x)
Wave Speed v = |b/a|
(coeff of t / coeff of x)
Tuning Fork Problems
Loading with wax DECREASES frequency. Filing INCREASES frequency. If beats decrease after loading, unknown fork had a HIGHER frequency.
Speed Hacks
Doppler mnemonic: 'TOP' - Towards Observer Positive. This sets the sign for vₒ in the numerator. The sign for vₛ in the denominator is always opposite.
For organ pipes, an open end is always an Antinode (A) and a closed end is always a Node (N). This instantly tells you the boundary conditions.
In standing waves on a string, the number of loops equals the harmonic number (n). 3 loops = 3rd harmonic.
If intensity changes from I₁ to I₂, the amplitude ratio is A₂/A₁ = √(I₂/I₁).
For reflected sound (echo) Doppler problems, apply the formula twice. First, the wall is the 'observer'. Second, the wall is the new 'source'.
Standing Waves: String vs. Pipes
| Property | String (Both ends fixed) | Open Pipe (Both ends open) | Closed Pipe (One end closed) |
|---|---|---|---|
Fundamental Freq (f₁) | v / 2L | v / 2L | v / 4L |
Harmonics Present | All (1, 2, 3, ...) | All (1, 2, 3, ...) | Only Odd (1, 3, 5, ...) |
nth Harmonic Freq (fₙ) | n * f₁ | n * f₁ | (2n-1) * f₁ |
Overtones | 1st = 2nd Harmonic | 1st = 2nd Harmonic | 1st = 3rd Harmonic |
End Conditions | Node - Node | Antinode - Antinode | Node - Antinode |