Stationary Waves

To calculate the wave speed on a stretched string, use the formula $v = \sqrt{\frac{T}{\mu}}$, where $T$ is the tension in Newtons and $\mu$ is the mass per unit length in $kg\,m^{-1}$.

The Fundamental Frequency ($f_0$), or first harmonic, occurs when the string length $L$ equals half a wavelength ($L = \frac{\lambda}{2}$). Substituting this into the wave equation $v = f\lambda$ gives $f_0 = \frac{1}{2L} \sqrt{\frac{T}{\mu}}$.

Higher harmonics are integer multiples of the fundamental frequency. For example, the second harmonic ($f_1$) has a frequency of $2f_0$ and the third harmonic ($f_2$) has a frequency of $3f_0$.

When analyzing air columns, identify if the pipe is open at both ends or closed at one end. Closed ends always form nodes, while open ends always form antinodes.

Check your units: Mass per unit length ($\mu$) is often given in grams per centimeter; always convert to $kg\,m^{-1}$ before using the wave speed formula.

Identify the Harmonic: Carefully count the number of 'loops' in a diagram. One loop is half a wavelength. If there are 3 loops, then $L = 1.5\lambda$.

Boundary Conditions: Remember that a fixed end (like a string tied to a post) MUST be a node, and a free end (like the open end of a tube) MUST be an antinode.

Sanity Check: If tension increases, the frequency should increase. If the string is made heavier (higher $\mu$), the frequency should decrease.

Energy Transfer: A common mistake is stating that stationary waves transfer energy. They do not; the energy 'sloshes' back and forth between potential and kinetic forms within the nodes.

Phase Difference: Students often think points on opposite sides of a node are in phase. In reality, points in adjacent loops are $180^{\circ}$ ($\pi$ radians) out of phase.

Wavelength vs Length: Do not confuse the length of the string ($L$) with the wavelength ($\lambda$). Always write out the relationship (e.g., $\lambda = 2L$ for the first harmonic) before calculating.