Waves and Oscillations

Essential formulas and concepts in waves and oscillations

Wave Speed

\(v = f \,\lambda\)

Variables:

Description/Usage:

Mathematical concept visualization

Period and Frequency

\(T = \frac{1}{f}\)

Variables:

  • \(T = period (time for one complete cycle), f = frequency.\)

Description/Usage:

Simply the reciprocal relationship between how long one cycle takes and how many cycles occur per second. If a pendulum swings back and forth in 2 seconds (\(T=2\) s), its frequency is \(0.5\) Hz.

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Pendulum (small angle) Period

\(T = 2\pi \sqrt{\frac{L}{g}}\)

Variables:

  • \(L = length of a simple pendulum (mass on a string), g = acceleration due to gravity. (This formula assumes small oscillation amplitude so that motion is approximately simple harmonic.)\)

Description/Usage:

Gives the time for one complete swing back-and-forth of a pendulum (for small angles). It shows period is independent of mass and amplitude (for small angles) and increases with the square root of length.

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Mass-Spring Oscillator Period

\(T = 2\pi \sqrt{\frac{m}{k}}\)

Variables:

  • \(m = mass attached to spring, k = spring force constant (stiffness).\)

Description/Usage:

The period of an object of mass \(m\) oscillating on a spring (assuming no damping). A stiffer spring (\(k\) large) gives a shorter period (faster oscillation), while a heavier mass gives a longer period.

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Wave Equation (one dimension)

\(\displaystyle \frac{\partial^2 y}{\partial t^2} = v^2\,\frac{\partial^2 y}{\partial x^2}\)

Variables:

  • \(y(x,t) describes the wave displacement as function of position x and time t. v = wave speed (assumed constant here).\)

Description/Usage:

A differential equation that \(y(x,t)\) satisfies for many physical waves (on a string, sound, light in 1D). It states that the acceleration of a small segment (left side) is proportional to the curvature of the wave shape (right side) with factor \(v^2\). Solutions are typically \(y(x,t)=f(x-vt)+g(x+vt)\) (waves traveling right and left).

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Sound Intensity Level (Decibels)

\(\displaystyle \beta = 10 \log_{10}\!\frac{I}{I_0}\,\text{dB}\)

Variables:

Description/Usage:

Converts a wide range of sound intensity into a logarithmic scale that correlates with perceived loudness. An increase of 10 dB means intensity is 10 times higher (approx. twice as loud to the human ear).

Mathematical concept visualization