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The Mathematics of Waves

The Mathematics of Waves. The Mathematics of Waves. As if there weren’t enough categories for types of waves yet… We can look at 2 different types of mathematical waves Pulse Waves Periodic Waves Much of this class will look at periodic waves. Pulse Waves.

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The Mathematics of Waves

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  1. The Mathematics of Waves

  2. The Mathematics of Waves • As if there weren’t enough categories for types of waves yet… • We can look at 2 different types of mathematical waves • Pulse Waves • Periodic Waves • Much of this class will look at periodic waves

  3. Pulse Waves • A pulse wave is when a single burst of energy is sent through a medium/space • Imagine having a straight rope and lifting it once to send a ‘pulse’ of energy through the rope

  4. Real Life Examples of pulse waves • If it were perfectly silent and someone blasted an air horn for a brief second. This would send a pulse of energy through the air molecules in the room. This would create a single pulse wave that travels through the air • In a perfectly dark room, if you flicked the lights on and then off for an instant. This would send a pulse of light through the room.

  5. Periodic Waves • Periodic waves are when continuous energy is sent through a medium/space • Imagine shaking the end of a rope up and down, over and over again

  6. The Mathematics of Waves • In a mechanical wave, the amplitude is related to the energy carried in the wave • The energy carried by the wave will be proportional to the amplitude squared • For Example - Think about creating waves in a rope. If you want to make the waves go higher, you have to work harder (put in more energy) • This is NOT NECESSARILYTHE CASE FOR ELECTROMAGNETIC WAVES

  7. Energy of a Wave • Example The amplitude of an ocean wave is 2.00 m high. If the wave doubles in height, by what factor does the energy carried by the wave increase?

  8. Velocity of a wave • Velocity – The velocity (as you learned in grade 10) is the distance travelled divided by the time • We know that with wave, the distance is measured in wavelengths and the time is measures in periods

  9. Velocity of a wave • We can re-write that equation such as: • We also know that 1/T is the frequency of a wave. So we can write:

  10. Velocity of a wave • So we now know that: • Velocity is in the units m/s

  11. Velocity of a wave • Example #1 Yellow light has a frequency of 508 THz. If the yellow light has a wavelength of 570 nm, what is the velocity of yellow light?

  12. Velocity of a wave • Example #2 The velocity of sound through air is 340 m/s. Middle C on a piano has a frequency of 261.6 Hz. What is the wavelength of middle C?

  13. Velocity of a wave • Example #3 Blue light has a frequency of 606 THz and a wavelength of 450.0 nm. How long would it take blue light to reach the Earth from the Sun if the Earth is 1.496*108 km away?

  14. What you should know • Pulse Wave • Periodic Wave • Crest • Trough • Amplitude • Amplitude and Energy • Period (T) • Frequency (f) • Wavelength (λ) • Velocity Formula

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