Waves on stretched strings are fascinating phenomena that depend on tension and linear density. Higher tension speeds up waves, while greater linear density slows them down. These factors affect wave speed, which is crucial for understanding string instruments and other vibrating systems.

The wave speed formula, v = √(T/μ), connects tension and linear density to wave behavior. This relationship helps explain how changing string properties alters frequency and wavelength. Understanding these concepts is key to grasping harmonic oscillations and standing waves in vibrating strings.

Wave Properties on a Stretched String

Factors affecting wave speed

• Tension
  • Force applied to the string that stretches it taut
  • Directly proportional to the square root of the tension ($v \propto \sqrt{T}$)
  • Higher tension leads to faster wave propagation (tightening a guitar string)
• Linear density
  • Mass per unit length of the string, denoted by the symbol $\mu$
  • Inversely proportional to the square root of the linear density ($v \propto \frac{1}{\sqrt{\mu}}$)
  • Lower linear density results in faster wave speed (thinner strings on a violin)

Wave speed calculation

• Mathematical relationship: $v = \sqrt{\frac{T}{\mu}}$
  • $v$ represents the wave speed measured in meters per second (m/s)
  • $T$ stands for the tension in the string, expressed in newtons (N)
  • $\mu$ symbolizes the linear density of the string in kilograms per meter (kg/m)
• Steps to calculate wave speed:
  1. Measure the tension applied to the string using a force sensor or spring scale
  2. Determine the linear density by dividing the string's mass by its length
  3. Plug in the values for tension and linear density into the wave speed equation
  4. Solve the equation for $v$ to obtain the wave speed

Effects of string properties

• Frequency and wavelength related by the equation: $v = f\lambda$
  • $f$ denotes the frequency of the wave in hertz (Hz)
  • $\lambda$ represents the wavelength measured in meters (m)
  • Altering string properties affects frequency and wavelength while maintaining this relationship
• Increasing tension
  • Raises the wave speed, causing the wavelength to increase if frequency remains constant (longer waves on a tighter string)
  • Increases the frequency if wavelength is held constant, resulting in higher pitch (tightening a violin string)
• Increasing linear density
  • Lowers the wave speed, leading to shorter wavelengths if frequency is unchanged (shorter waves on a thicker string)
  • Decreases the frequency when wavelength remains constant, producing a lower pitch (using a thicker guitar string)
• Amplitude of the wave does not affect its speed on a stretched string

Harmonic oscillations and standing waves

• Harmonic oscillation occurs when the string vibrates at its natural frequencies
• Standing waves form when waves traveling in opposite directions interfere constructively
  • Nodes: points of minimum amplitude
  • Antinodes: points of maximum amplitude
• Resonance occurs when the driving frequency matches a natural frequency of the string
  • Results in increased amplitude of vibration