Wave Speed on a Stretched String

• Wave Speed ($v$): The velocity at which a transverse pulse or wave travels along a medium, measured in meters per second ($m \cdot s^{-1}$). In a string, this speed is constant for a given tension and material density.

• Tension ($T$): The pulling force exerted by the string, typically measured in Newtons (N). Higher tension increases the restoring force, causing the string to return to equilibrium faster and thus increasing wave speed.

• Mass per Unit Length ($\mu$): Also known as linear density, it represents the mass of the string divided by its total length ($\mu = \frac{m}{L}$), measured in $kg \cdot m^{-1}$. A 'heavier' string (higher $\mu$) has more inertia, which resists acceleration and slows down the wave.

• The speed of a wave on a string is mathematically derived from the balance of tension and inertia. The governing formula is:

$v = \sqrt{\frac{T}{\mu}}$

• This formula indicates that wave speed is proportional to the square root of tension and inversely proportional to the square root of the linear density.

• When combined with the general wave equation $v = f\lambda$, we can determine the fundamental frequency ($f_0$) of a string fixed at both ends. Since the first harmonic has a wavelength $\lambda = 2L$, the frequency is given by:

$f = \frac{1}{2L} \sqrt{\frac{T}{\mu}}$

• Calculating Linear Density: To find $\mu$, measure the total mass of a long sample of the string using a high-precision balance and divide by its total length. It is more accurate to measure a long piece (e.g., 2 meters) to reduce percentage uncertainty.

• Determining Tension: In laboratory settings, tension is usually provided by hanging masses over a pulley. Calculate tension using $T = mg$, where $m$ is the hanging mass and $g$ is the gravitational field strength ($9.81 \, N \cdot kg^{-1}$).

• Graphical Analysis: To verify the relationship experimentally, plot frequency ($f$) on the y-axis against the inverse of length ($1/L$) on the x-axis. The gradient of this linear graph represents $\frac{v}{2}$, allowing for the calculation of wave speed.

• Length vs. Wavelength: It is critical to distinguish between the physical length of the string ($L$) and the wavelength ($\lambda$). For the first harmonic, $\lambda = 2L$; for the second, $\lambda = L$; and for the third, $\lambda = \frac{2}{3}L$.

• Unit Consistency: Always convert mass to kilograms ($kg$) and length to meters ($m$) before calculating $\mu$. Strings are often measured in grams or centimeters, which will lead to incorrect orders of magnitude if not converted.

• The Square Root Trap: Remember that doubling the tension does NOT double the wave speed; it increases the speed by a factor of $\sqrt{2} \approx 1.41$.

• Vibrating Length: When a string is fixed between two bridges, use the distance between the bridges as $L$ for frequency calculations, even if the physical string is longer.

• Sanity Check: Wave speeds on typical metal or nylon strings usually range from $50$ to $200 \, m \cdot s^{-1}$. If your calculation results in a speed near the speed of light or slower than a walking pace, re-check your units.