Study Notes on Waves and Stationary Waves

Calculation of Beat Frequency

  • To find the beat frequency (denoted as $f_b$):
    • The formula used is:
      f<em>b=∣f</em>1−f2∣f<em>b = |f</em>1 - f_2|
    • Substituting values:
    • $f_1 = 1750 Hz$
    • $f_2 = 3900 Hz$
    • Calculation gives:
      fb=∣1750−3900∣=2150Hzf_b = |1750 - 3900| = 2150 Hz
    • Note: These beats cannot be detected as separate by the human ear because they fall outside the frequency range that humans can discern (roughly 20 Hz to 20 kHz).

Stationary Waves (or Standing Waves)

  • Definition:
    • Standing waves do not appear to move; they vibrate around a mean position.
    • Example: Circular standing waves on a surface of liquid (e.g., milk) caused by vibrations from an external source like a refrigerator motor.
  • Formation of Standing Waves:
    • Occurs when two identical waves with the same speed, amplitude, and frequency travel in opposite directions and are superposed.
    • Reflection from boundaries creates constructive and destructive interference, leading to stationary waves.
  • Diagram (Figure 9.7):
    • Seven positions demonstrate displacement over time:
    • At $t = 0, rac{1}{4}T, rac{1}{2}T, rac{3}{4}T, T$
    • Points 1, 3, 5, and 7 do not vibrate (nodes).
    • Points 2, 4, and 6 oscillate maximally (antinodes) with amplitude equal to the sum of the two component waves’ amplitudes.
  • Characteristics of Stationary Waves:
    1. No energy is transferred between particles.
    2. Distance between successive nodes or antinodes is half the wavelength ($ rac{ ext{A}}{2}$).
    3. Distance from node to antinode is a quarter of the wavelength ($ rac{ ext{A}}{4}$).

Stationary Waves in a Stretched String

  • Stationary waves also present in strings of musical instruments due to wave reflections from the string ends.

Modes of Vibration in a Stretched String

  1. Frequency Calculation:
    • Setup: Length of string ($L$) leads to tension ($T$).
  2. Loop Formation:
    • Single Loop (Fundamental Frequency $f_1$):
      • String plucked at the middle:
      • L=12extA1L = \frac{1}{2} ext{A}_1
      • extA1=2Lext{A}_1 = 2L
      • $v = f1 ext{A}1$:
        • f<em>1=vextA</em>1=T2Lext×MLf<em>1 = \frac{v}{ ext{A}</em>1} = \frac{T}{2L} ext{×}\frac{M}{L}
        • Resulting in:
          f1=12LTextMf_1 = \frac{1}{2L}\frac{T}{ ext{M}}
    • Two Loops (Second Harmonic $f_2$):
      • Plucked at a quarter length:
      • L=22extA<em>2L = \frac{2}{2} ext{A}<em>2; hence f</em>2=2f1f</em>2 = 2f_1
    • Three Loops (Third Harmonic $f_3$):
      • Plucked at one sixth length:
      • L=32extA3L = \frac{3}{2} ext{A}_3
      • Resulting in: f<em>3=3f</em>1f<em>3 = 3f</em>1
  3. General Formulae for n Loops:
    • f<em>n=nf</em>1f<em>n = n f</em>1
    • L=nextA2L = \frac{n ext{A}}{2}
    • Quantization of frequencies occurs due to resonance.

Stationary Waves in an Air Column

  • Standing waves occur in air columns, such as in musical instruments (organ pipes, flutes).
  1. Types of Organ Pipes:
    • Open: Both ends open to air.
    • Closed: One end closed and one end open.
  2. Wave Reflection:
    • Reflection from the closed end of a tube produces nodes (minimum movement) and antinodes (maximum movement).
  3. Vibration and Resonance:
    • Reflected wave creates a standing wave where the length of the tube relates to wavelength:
      • L=14extA1L = \frac{1}{4} ext{A}_1

Modes of Vibration in Open and Closed Pipes

Open Pipe:
  1. Fundamental Frequency:
    • L=12extAL = \frac{1}{2} ext{A} leading to f1=v2Lf_1 = \frac{v}{2L}.
  2. Second Harmonic:
    • L=extA<em>2:f</em>2=vL=2f1L = ext{A}<em>2 : f</em>2 = \frac{v}{L} = 2f_1
  3. Third Harmonic:
    • L=32extA<em>3:f</em>3=3f1L = \frac{3}{2} ext{A}<em>3 : f</em>3 = 3f_1
Closed Pipe:
  1. Fundamental Frequency:
    • L=14extA:f1=v4LL = \frac{1}{4} ext{A} : f_1 = \frac{v}{4L}.
  2. Second Harmonic:
    • Exists only in odd harmonics, leading to frequency expressions like fn=(2n−1)v4Lfn = (2n-1)\frac{v}{4L}.

Real-world Applications and Phenomena

  • The concept of standing waves is relevant in understanding structural integrity during earthquakes. - Buildings of specific heights resonate with seismic waves, signifying why some buildings collapse while others remain intact. - Reflective qualities of waves lead to constructive and destructive interference, affecting the stability of structures during seismic activity.

Experimental Demonstration of Standing Waves

  • Suggested experiment: Place a bowl of milk on a fan to observe circular standing waves created at the surface due to vibrations.