Standing Waves: Quick Revision

Standing waves, also known as stationary waves, arise from the superposition of two waves traveling in opposite directions with the same frequency, amplitude, and wavelength. Instead of propagating, the wave appears to oscillate in place.

Formation

• Superposition Principle: When two waves meet, the resultant displacement at any point is the algebraic sum of the displacements of the individual waves.
• Reflection: Standing waves are commonly formed when a traveling wave reflects off a boundary and interferes with the incident wave.

Key Characteristics

• Nodes: Points along the medium where the amplitude of the wave is always zero. These occur at points of destructive interference. The distance between adjacent nodes is $\frac{\lambda}{2}$.
• Antinodes: Points along the medium where the amplitude of the wave is maximum. These occur at points of constructive interference. The distance between adjacent antinodes is also $\frac{\lambda}{2}$.
• Wavelength ($\lambda$) and Frequency ($f$): These remain the same as the original traveling waves.
• No Net Energy Transfer: Unlike traveling waves that transfer energy from one point to another, standing waves trap energy within the loops formed between nodes.
• Quantized Frequencies (for bounded media): When standing waves are formed in a medium with fixed boundaries (like a string fixed at both ends or a closed pipe), only certain frequencies can produce stable standing wave patterns. These are called resonant frequencies or harmonics.

Standing Waves on a String Fixed at Both Ends

• Boundary Conditions: The displacement at both ends must always be zero (nodes).
• Allowed Wavelengths: The length of the string ($L$) must be an integer multiple of half the wavelength: $L = n \frac{\lambda_n}{2} \implies \lambda_n = \frac{2L}{n}$ where $n = 1, 2, 3, ...$ is the harmonic number.
• Allowed Frequencies (Harmonics): Using the wave speed $v = f\lambda$, the resonant frequencies are: $f_n = \frac{v}{\lambda_n} = n \frac{v}{2L}$
  • $n=1$: Fundamental frequency (first harmonic)
  • $n=2$: First overtone (second harmonic)
  • $n=3$: Second overtone (third harmonic), and so on.

Standing Waves in Pipes

The boundary conditions at the ends of the pipe (open or closed) determine the allowed wavelengths and frequencies.

Closed at One End, Open at the Other:

• Boundary Conditions: Node at the closed end, antinode at the open end.
• Allowed Wavelengths: $L = (2n - 1) \frac{\lambda_n}{4} \implies \lambda_n = \frac{4L}{2n - 1}$ where $n = 1, 2, 3, ...$
• Allowed Frequencies (Odd Harmonics Only): $f_n = \frac{v}{\lambda_n} = (2n - 1) \frac{v}{4L}$
  • $n=1$: Fundamental frequency (first harmonic)
  • $n=2$: First overtone (third harmonic)
  • $n=3$: Second overtone (fifth harmonic), and so on.

Open at Both Ends:

• Boundary Conditions: Antinodes at both ends.
• Allowed Wavelengths: $L = n \frac{\lambda_n}{2} \implies \lambda_n = \frac{2L}{n}$ where $n = 1, 2, 3, ...$
• Allowed Frequencies (All Harmonics): $f_n = \frac{v}{\lambda_n} = n \frac{v}{2L}$ This is the same formula as for a string fixed at both ends.

Key Formulas to Remember

• Distance between adjacent nodes/antinodes: $\frac{\lambda}{2}$
• Standing waves on a string/pipe open at both ends: $L = n \frac{\lambda}{2}$, $f_n = n \frac{v}{2L}$
• Standing waves in a pipe closed at one end: $L = (2n - 1) \frac{\lambda}{4}$, $f_n = (2n - 1) \frac{v}{4L}$
• Wave speed: $v = f\lambda$

Understanding the conditions for node and antinode formation at boundaries is crucial for determining the allowed modes of vibration in different systems.