Periodic Waves

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Discussion

speed, amplitude, wavelength, period, phase, frequency

seismic energy propagates through the earth within the 50 Hz to 0.001 Hz range! The expression "long period" (greater than 10 seconds) and "short period" (less than 1 second or greater than 1 Hz)

Summary

• Fundamentals
  • A wave is a disturbance that propagates.
  • A cycle is a sequence of events that repeats.
  • A cycle is periodic if it repeats with a characteristic time.
• Characteristics
  • Amplitude (A) is the maximum absolute value of a periodically varying quantity.
    • Amplitude has the unit of the quantity that is changing (ex. displacement, pressure, field strength, etc.)
  • Period (T) is the time between successive cycles of a repeating sequence of events.
    • T = t/n (time per number of cycles)
    • The SI unit of period is the second [s].
  • Frequency (ƒ) is the number of cycles of a repeating sequence of events in a unit interval of time.
    • ƒ = n/t (number of cycles per time)
    • Frequency and period are reciprocals (or inverses) of one another: ƒ = 1/T.
    • The SI unit of frequency is the hertz [Hz = 1/s = s⁻¹].
  • Phase (ϕ) is the stage of development of a periodic process.
    • Two points on a wave with the same phase have the same …
      • quantity of disturbance (ex. displacement) and
      • rate of change of disturbance (ex. velocity).
    • Phase is an angular quantity.
      • Adjacent points in phase are separated by one complete cycle.
      • Adjacent points out of phase are separated by half a cycle.
    • The SI unit of phase is the radian, which is itself a unitless ratio [rad = m/m = Pa/Pa = (V/m)/(V/m) = etc.].
  • Wavelength (λ) is the distance between any point on a periodic wave and the next point corresponding to the same portion of the wave measured along the path of propagation.
    • Wavelength is measured between adjacent points in phase.
    • The SI unit of wavelength is the meter [m].
  • Speed (v) is …
    • v = Δs/Δt the rate of change of distance with time by definition and
    • v = λƒ the product of wavelength and frequency for periodic waves.
    • Frequency and wavelength are inversely proportional.
      • Lower frequency waves have longer wavelengths.
      • Higher frequency waves have shorter wavelengths.
    • The speed of a wave is sometimes known as its wave speed
    • The SI unit of speed is the meter per second [m/s].
• One-Dimensional Wave Equation
  • ƒ (x, t) = A sin (2π (ft − x/λ) + ϕ)
    • A, amplitude
    • ƒ, frequency
    • λ, wavelength
    • ϕ, phase
  • ƒ (x, t) = A sin (ωt − kx + ϕ)
    • A, amplitude
    • ω, angular frequency, ω = 2πf
    • k, wave number k = 2π/λ
    • ϕ, phase

Problems

practice

1. The graph below is a record of sea level heights recorded at Hanimaadhoo, Maldives (6° 46' N 73° 10' E) during the tsunami of 26 December 2004. The data were filtered to eliminate the normal tidal fluctuations, so what you are seeing is the increase in sea level due to the tsunami (Data Source: University of Hawaii Sea Level Center).

   
   [magnify]

   Natural phenomena are normally very noisy (in the statistical sense) but from 10:50 to 12:05 local time the changes in sea level at Hanimaadhoo were most nearly periodic. During this time interval determine the tsunami's mean …
   1. amplitude and
   2. period
   The speed of a tsunami varies with depth. In the open ocean they normally move as fast as a commercial jet airplane (about 250 m/s or 900 kph) but slow down to the speed of a car on a neighborhood street when they reach the shallow waters of the shore (about 15 m/s or 55 kph). Given these speeds, determine the mean wavelength of the segment of the tsunami that arrived in Hanimaadhoo between 10:50 and 12:05 when they were …
   3. near the shore and
   4. in deep water
   One final question.
   5. How would the amplitude of a tsunami near shore compare to the amplitude of the same wave in the open ocean? Explain your reasoning.
   • Answer it.
2. Write something else.
   • Answer it.
3. Write something different.
   • Answer it.
4. Write something completely different.
   • Answer it.

numerical

1. AM radio broadcasts in the US are allocated frequencies from 530 KHz to 1710 KHz. What range of wavelengths correspond to these frequencies?
2. FM radio broadcasts in the US are allocated frequencies from 88 MHz to 108 MHz. What range of wavelengths correspond to these frequencies?
3. Visible light ranges from about 400 nm to about 700 nm. What range of frequencies correspond to these wavelengths?

Resources

• general
  • Longitudinal and Transverse Waves, Dan Russell, Kettering University [excellent animations of wave types]

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