Waves – Complete Formula Revision

Waves – Complete Formula Revision          (JEE Mains)

Every Formula | Every Trap | PYQ Mapping

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1️⃣ Basic Wave Parameters

Wave equation:

\[ y(x,t) = A \sin(kx - \omega t + \phi) \]

• Amplitude = \( A \)
• Angular frequency = \( \omega = 2\pi f \)
• Wave number = \( k = \frac{2\pi}{\lambda} \)
• Wave speed = \( v = \frac{\omega}{k} \)
• Also: \( v = f\lambda \)

⚠ Trap: Sign in wave equation decides direction.

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2️⃣ Speed of Transverse Wave on String

\[ v = \sqrt{\frac{T}{\mu}} \]

• T = tension
• \( \mu = \frac{m}{L} \) (linear density)

⚠ Increasing tension increases speed, but frequency remains constant.

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3️⃣ Longitudinal Sound Waves

Speed in medium:

\[ v = \sqrt{\frac{B}{\rho}} \]

• B = Bulk modulus
• \( \rho \) = density

For gas:

\[ v = \sqrt{\frac{\gamma P}{\rho}} \]

⚠ Trap: Speed independent of frequency.

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4️⃣ Principle of Superposition

Resultant displacement: \[ y = y_1 + y_2 \] Resultant amplitude: \[ A = \sqrt{A_1^2 + A_2^2 + 2A_1A_2 \cos\phi} \] Special cases:

• Constructive → \( A = A_1 + A_2 \)
• Destructive → \( A = |A_1 - A_2| \)

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5️⃣ Standing Waves

For string fixed at both ends: \[ \lambda_n = \frac{2L}{n} \] \[ f_n = \frac{nv}{2L} \] For one end closed pipe: \[ f_n = \frac{nv}{4L} \quad (n=1,3,5...) \] For open pipe: \[ f_n = \frac{nv}{2L} \]

⚠ Closed pipe has only odd harmonics.

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6️⃣ Beats

\[ f_{beat} = |f_1 - f_2| \]

Used for tuning instruments.

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7️⃣ Doppler Effect

\[ f' = f \left(\frac{v \pm v_o}{v \mp v_s}\right) \]

• Use + when moving towards
• Use − when moving away

⚠ Most common JEE mistake: Wrong sign selection.

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8️⃣ Energy & Intensity

Energy ∝ \( A^2 \) Intensity: \[ I = \frac{1}{2} \rho v \omega^2 A^2 \]

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🎯 JEE Mains Question Profiling

• Wave equation identification
• Speed change when tension changes
• Pipe harmonic problems
• Doppler sign trap
• Beat frequency numericals
• Amplitude-phase resultant formula

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✔ Waves = Formula + Sign Discipline + Harmonic Logic

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