Wave Speed on a Stretched String

• Fundamental Relationship: The speed of a transverse wave on a stretched string is directly proportional to the square root of the tension and inversely proportional to the square root of the linear mass density. This means increasing tension makes waves travel faster, while increasing the string's 'heaviness' (linear mass density) makes them travel slower.

• Mathematical Formulation: The relationship is encapsulated by the formula:

$v = \sqrt{\frac{T}{\mu}}$ Here, $v$ is the wave speed in meters per second (m s$^{-1}$), $T$ is the tension in Newtons (N), and $\mu$ is the linear mass density in kilograms per meter (kg m$^{-1}$). This formula is derived from considering the forces acting on a small segment of the string as a wave passes through it.

• Physical Intuition: Higher tension provides a stronger restoring force, allowing the disturbance to propagate more rapidly. Conversely, a higher linear mass density means more mass needs to be accelerated for the wave to move, leading to a slower propagation speed due to increased inertia.

• Definition: Linear mass density ($\mu$) is defined as the total mass ($m$) of the string divided by its total length ($L_{string}$). It is a crucial property that quantifies how 'heavy' a string is per unit of its length.

• Formula:

$\mu = \frac{m}{L_{string}}$ For accurate calculations, the mass $m$ must be in kilograms (kg) and the total length of the string $L_{string}$ must be in meters (m), resulting in $\mu$ in kg m$^{-1}$. It is important to use the total length of the string being weighed, not necessarily the vibrating length.

• Practical Measurement: To determine $\mu$, one typically measures the mass of a known length of the string using a balance and a ruler. For example, if a 1-meter segment of string has a mass of 0.005 kg, then $\mu = 0.005 \text{ kg/m}$.

• Wave Equation: The wave speed $v$ on the string is also related to the frequency ($f$) and wavelength ($\lambda$) of the wave by the general wave equation: $v = f\lambda$. This equation applies to both traveling and stationary waves.

• Stationary Waves on Fixed Strings: When a string is fixed at both ends, only specific wavelengths can form stationary waves, known as harmonics. The simplest stationary wave, the fundamental frequency (or first harmonic), corresponds to a single loop where the string length ($L$) is exactly half a wavelength, meaning $\lambda = 2L$.

• Fundamental Frequency Formula: By combining the wave speed formula ($v = \sqrt{\frac{T}{\mu}}$) with the wave equation ($v = f\lambda$) and the condition for the fundamental frequency ($\lambda = 2L$), we can derive the formula for the fundamental frequency ($f_0$):

$f_0 = \frac{1}{2L} \sqrt{\frac{T}{\mu}}$ This formula shows that the fundamental frequency is inversely proportional to the string's length and directly proportional to the square root of tension and inversely proportional to the square root of linear mass density.

• Tension ($T$): Increasing the tension in a string significantly increases the wave speed. Since $v \propto \sqrt{T}$, quadrupling the tension will double the wave speed. This is why tightening a guitar string raises its pitch, as higher wave speed leads to higher frequency for a given wavelength.

• Linear Mass Density ($\mu$): Increasing the linear mass density of a string decreases the wave speed. Since $v \propto \frac{1}{\sqrt{\mu}}$, using a string that is four times heavier per unit length will halve the wave speed. This explains why bass guitar strings are thicker (higher $\mu$) than treble strings, producing lower frequencies.

• Length ($L$): While string length does not directly affect the wave speed itself (which is an intrinsic property of the string material and tension), it critically determines the possible wavelengths and thus the frequencies of stationary waves that can be formed. For a given wave speed, a longer string will support longer wavelengths and therefore lower frequencies.

• Unit Conversion is Critical: Always ensure all quantities are in their standard SI units before performing calculations. Tension must be in Newtons (N), mass in kilograms (kg), and lengths in meters (m). Forgetting to convert grams to kilograms or centimeters to meters is a very common error.

• Identify the Correct Length: Distinguish between the total length of the string used to calculate linear mass density ($\mu$) and the vibrating length ($L$) of the string between fixed points, which determines the wavelength of stationary waves. These two lengths may not always be the same.

• Step-by-Step Calculation: When solving problems involving wave speed and frequency, break down the calculation into logical steps: first calculate $\mu$, then $v$, and finally $f$ or $\lambda$ as required. This reduces errors and makes your working clear for partial credit.