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School-college Physics Notes: Waves 10. Wave equation calculations

Introduction to waves: 10. WAVE CALCULATIONS - formulae and how to solve wave problem questions

Doc Brown's Physics exam study revision notes

10. WAVE CALCULATIONS - formulae and how to solve wave problem questions

• Be able to use both the equations below, which apply to all waves (and their rearrangements):

• appropriate units used in ()

• a) wave speed (metre/second, m/s) = frequency (hertz, Hz, s-1) x wavelength (metre, m)

• in 'shorthand'    v = f x λ

• rearrangements:

• frequency = speed ÷ wavelength,   f = v ÷ λ

• wavelength = speed ÷ frequency,   λ = v ÷ f

• b) wave speed (metre/second, m/s) = distance (metre, m) ÷ time (second, s)

• in 'shorthand'   v = d ÷ t

• rearrangements:  d = v x t   and   t = d ÷ v

• This is the general formula for the speed or velocity of anything moving.

• (a) , (b) • Note that you are not required to recall the value of the speed of electromagnetic waves through a vacuum ...

• .. it is very big, 'speed of light' = v = 3 x 108 m/s

• Be able to do examples of calculations using the wave speed formula and its rearrangements.

• Wave frequencies are often given in hertz (Hz), kilohertz (kHz) or megahertz (MHz).

• 1 kHz   1000 Hz,  1 MHz 1 000 000 Hz, 1 MHz 1000 kHz

Examples of wave calculations

Q1 A wave has a speed of 0.25 m/s and a wavelength of 5.0 cm.

(a) Calculate the frequency of the wave

(b) Calculate the period of the wave.

Worked out ANSWERS to wave calculation questions

Q2 The frequency of sound in air at room temperature and pressure is 343 m/s.

The musical note middle C has a frequency of 262 Hz.

Calculate the wavelength of the middle C sound.

Worked out ANSWERS to wave calculation questions

Q3 A set of ocean waves has a frequency of 0.50 Hz.

If the average distance between the crests of the waves is 10 m, what is the average speed of the ocean waves?

Worked out ANSWERS to wave calculation questions

Q4 The time period of a radio electromagnetic wave is 5.0 x 10-5 seconds

(a) What is the frequency of the radio wave?

(b) If the 'speed of light' is 3.0 x 108 m/s, calculate the wavelength of the radio wave in m.

(c) If the distance from the radio station to your radio is 200 km, how long does it take the signal to reach you?

Worked out ANSWERS to wave calculation questions

Q5 A water wave has a frequency of 0.50 Hz and a wavelength of 150 cm.

(a) Calculate the speed of the wave in m/s.

(b) If the frequency of this wave triples, what will be its wavelength? and what assumption have you made?

(c) If the frequency of the original wave doubles, and the wavelength of the wave quadruples, what will be the new speed of the wave?

Worked out ANSWERS to wave calculation questions

Q6 Suppose an airliner sends out a microwave radar signal of wavelength of 1.20 cm.

The microwave reflects off another aircraft and the echo is detected after a time lapse of 6.0 µs.

The speed of electromagnetic radiation = 3.00 x 108 m/s.

(a) What is the frequency of the microwave beam?

Worked out ANSWERS to wave calculation questions

(b) What is the distance between the two aircraft?

Worked out ANSWERS to wave calculation questions

Q7 A satellite is 75 km above the Earth's surface. (speed of light 3.00 x 108 m/s)

What is the shortest time that a microwave signal would take to reach the satellite from the Earth's surface?

Worked out ANSWERS to wave calculation questions

Q8 A red light wave has a wavelength of 7.0 x 10-7 m. (speed of light 3.00 x 108 m/s)

What is the frequency of the light wave?

Worked out ANSWERS to wave calculation questions

Need more varied questions?

Keywords, phrases and learning objectives for waves

Be able to use the wave equation formula to perform calculation using the correct appropriate units.

Know how to solve wave problem questions in a variety of situations and for different types of waves.

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Worked out ANSWERS to the wave calculation questions

Q1 A wave has a speed of 0.25 m/s and a wavelength of 5.0 cm.

(a) Calculate the frequency of the wave

f = v ÷ λ

5.0 cm 5/100 m = 0.05 m

Therefore frequency = 0.25/0.05 = 5.0 Hz

(b) Calculate the period of the wave.

period = 1 / frequency = 1 / 5 = 0.20 s

Q2 The frequency of sound in air at room temperature and pressure is 343 m/s.

The musical note middle C has a frequency of 262 Hz.

Calculate the wavelength of the middle C sound.

λ = v ÷ f

wavelength = 343 ÷ 262 = 1.31 m (131 cm, 3 s.f.)

Q3 A set of ocean waves has a frequency of 0.50 Hz.

If the average distance between the crests of the waves is 10 m, what is the average speed of the ocean waves?

v = f x λ

speed (m/s) = frequency (Hz) x wavelength (m)

speed = 0.5 x 10 = 5.0 m/s

Q4 The time period of a radio electromagnetic wave is 5.0 x 10-5 seconds

(a) What is the frequency of the radio wave?

period = 1 / frequency

f = 1 / period = 1 / (5.0 x 10-5) = 2.0 x 104 Hz

(b) If the 'speed of light' is 3.0 x 108 m/s, calculate the wavelength of the radio wave in m.

λ = v ÷ f = 3.0 x 108 / 2 x 104 = 1.5 x 104 m

(c) If the distance from the radio station to your radio is 200 km, how long does it take the signal to reach you?

The speed formula is v = d / t,  so  t = d / v = (200 x 1000) / (3.0 x 108) = 6.7 x 10-4 s

Q5 A water wave has a frequency of 0.50 Hz and a wavelength of 150 cm.

(a) Calculate the speed of the wave in m/s.

v = f x λ = 0.50 x (150/100) = 0.075 m/s

(b) If the frequency of this wave triples, what will be its wavelength? and what assumption have you made?

If you assume the speed stays the same, the wavelength will be a third of 150 cm, 50 cm or 0.50 m,

because frequency x wavelength (f x λ) is a constant for a constant speed.

(c) If the frequency of the original wave doubles, and the wavelength of the wave quadruples, what will be the new speed of the wave?

v = f x λ, putting in the factors gives 2 x (1/4) = 0.5, so the new speed will be 0.5 x 0.075 = 0.038 m/s  (2 sf)

Q6 Suppose an airliner sends out a microwave radar signal of wavelength of 1.20 cm.

The microwave reflects off another aircraft and the echo is detected after a time lapse of 6.0 µs.

The speed of electromagnetic radiation = 3.00 x 108 m/s.

(a) What is the frequency of the microwave beam?

speed = wavelength x frequency

f = v ÷ λ = 3.00 x 108 ÷ (1.20 / 100) = 2.50 x 1010 Hz

(b) What is the distance between the two aircraft?

s = d / t,  d = s x t = 3.00 x 108 x 6.0 x 10-6 = 1800 m (total distance including echo)

distance between aircraft = 1800 ÷ 2 = 900 m

(µ is micro = 10-6, and total distance is halved because it involves 'there and back')

Q7 A satellite is 75 km above the Earth's surface. (speed of light 3.00 x 108 m/s)

What is the shortest time that a microwave signal would take to reach the satellite from the Earth's surface?

s = d / t,  t = d / s = (75 x 1000) / (3.00 x 108) = 2.50 x 10-4 seconds

Q8 A red light wave has a wavelength of 7.0 x 10-7 m. (speed of light 3.00 x 108 m/s)

What is the frequency of the light wave?

v = f x λ  *  f = v / λ  = 3 x 108 / (7.0 x 10-7 m) = 4.29 x 1014 Hz  (3 sf)

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