Wave Equations

The relationship between speed, frequency, and wavelength is governed by the equation: $v = f \lambda$

This equation states that the speed of a wave is the product of its frequency and its wavelength. Because the speed of a wave is often constant in a specific medium, frequency and wavelength are inversely proportional; as frequency increases, wavelength must decrease.

For non-periodic motion or measuring average speed over a distance, the general motion equation is used: $v = \frac{d}{t}$

In this context, $d$ is the total distance the wave has traveled and $t$ is the time taken. This is particularly useful in experimental settings where a wave pulse is timed over a known distance.

Variable Isolation: To find frequency, rearrange the formula to $f = \frac{v}{\lambda}$. To find wavelength, use $\lambda = \frac{v}{f}$.

Unit Standardization: Always ensure units are in the SI standard before calculating. Convert centimeters to meters ($100 cm = 1 m$) and kilohertz to hertz ($1 kHz = 1000 Hz$).

Step-by-Step Calculation: First, identify the known values from the problem. Second, convert all units to meters and hertz. Third, select the appropriate form of the wave equation. Finally, substitute and solve.

Wave Speed vs. Frequency: Wave speed is determined by the medium (e.g., sound travels faster in water than in air), whereas frequency is determined by the source of the vibration.

Period vs. Frequency: While frequency is cycles per second, the period ($T$) is the time taken for one complete cycle ($T = 1/f$).

Check Unit Consistency: Examiners often provide wavelength in centimeters ($cm$) or millimeters ($mm$) to test your ability to convert to meters ($m$). Always perform this conversion first.

The 'Reasonableness' Check: For sound waves in air, speed is roughly $330-340 m/s$. If your calculation for a sound wave results in $3,000,000 m/s$, re-check your decimal places or unit conversions.

Formula Selection: Use $v = d/t$ when given a total distance and time (like an echo or a wave traveling across a pool). Use $v = f \lambda$ when given periodic properties like frequency or wavelength.

Prefix Awareness: Be alert for prefixes like 'milli' ($10^{-3}$), 'centi' ($10^{-2}$), and 'kilo' ($10^3$). A frequency of $2 kHz$ must be entered as $2000 Hz$ in the equation.

Confusing $v$ and $f$: Students sometimes mistake the speed of the wave for the frequency of the vibration. Remember that speed is how fast the energy moves forward, while frequency is how fast the source wiggles.

Incorrect Wavelength Measurement: When looking at a diagram, ensure the wavelength is measured between two identical points. Measuring from a peak to a trough only gives half a wavelength ($0.5 \lambda$).

Ignoring the Medium: A common mistake is assuming wave speed changes when frequency changes. In a uniform medium, the speed remains constant; only the wavelength adjusts to compensate for frequency changes.