Wave Velocity Exercises
Tie one end of a rope to a wall, shake the other end up and down, and you get a wave travelling along the rope. The velocity of the wave v, the frequency f, and the wavelength λ, are related by
In the case of the rope wave, v could be the velocity of a crest on the wave, f is the frequency that a point on the rope moves up and down, and the distance between two adjacent crests is the wavelength. These quantities are simple enough to visualise.
What about water wave, sound wave or radio wave?
Water wave
In water wave, a crest is a highest point on the surface wave. Such points can form a line, like the circle in a ripple. This line is a wavefront, as is any line joinings points of the same height. So a line joining the troughs would also be a wavefront. The spacing between two wavefronts is the wavelength. The wave velocity is the velocity of a crest along a line that is perpendicular to the wavefront. The frequency is the number of times that a point on the water surface moves up and down in one second.
Exercise 1. Drop a stone in a pond. Suppose the ripples travels outward at a velocity of 0.1 m/s, and the spacing between two adjacent ripples is 2 cm. How many times does a point on the water surface move up and down every second?
Answer. The wavelength λ is 0.02 m and the velocity v is 0.1 m/s. The number of times that a point moves up and down is the frequency f. Using the above formula, the frequency is
Sound wave
Sound wave is a pressure wave through the air. The air gets alternately compressed and rarefied (the opposite of compressed). If you think of a fixed point in the air as the sound travels over it, the pressure there would keep going up and down. The number of times per second that this happens is the frequency. If you can freeze time and move along the sound direction, the pressure would go up and down. A point with the highest pressure is the crest, and the high pressure region forms a surface. There many of such parallel surfaces as we move away from the source of the sound. These are the wavefronts.
The spacing between two adjacent wavefronts is the wavelength. Now let time move again. The velocity of a crest in a direction perpendicular of the wavefront is the wave velocity. (The pressure change in a sound wave is normally very small, unless the sound is very loud.)
Exercise 2. If you hit the A note key on the piano, you hear a sound with a frequency of 440 Hz. The speed of sound in air is 330 m/s. What is the distance between two high pressure regions in the air?
Answer. This is a very vague question. There are so many high pressure regions in a sound wave - which ones are we talking about? The technique is think for a few possible examples, and look for a answer that is meaningful. This is still vague, but lets try. Now the high pressure regions are the crests. These form surfaces that we call wavefronts. To find a distance, we need two points. First, try two points on the same wavefront. These two points can be touching, or they can be anywhere on the surface. So the distance can be zero to any value. Not very helpful. Next, think of two points on adjacent wavefronts. The two points can again be anywhere, so there are many possible distances. At their closest, the distance happens to be the wavelength. Now this has meaning - in the sense that it relates to something we know. So I would say that for this question, we need to find the wavelength. Using the above formula again, the answer is
Radio wave is produced by electric current oscillating in a wire. This produces an oscillating magnetic field around it. A changing magnetic field induces an electric field, which in turn generates a magnetic field further away. The magnetic field is always perpendicular to the electric field. In this way, an electromagnetic wave is produced. This makes up the radio wave.
As this wave passes over a point, the electric field at that point oscillates in strength and direction. The number of oscillations per second is the frequency. If we freeze time and look along the direction of the wave, we would see the electric field increase and decrease in strength. The point of maximum field is the crest, and a surface containing the crests is the wavefront. There are many such parallel wavefronts.
The spacing between the adjacent wavefronts (of crests) is the wavelength. If we allow time to flow again, the velocity of a crest, along a line perpendicular to a wavefront, is the velocity of the radio wave.
Exercise 3. The radio wave from a radio station produces an electric field that oscillates at a frequency of 100 MHz at a radio antenna. The velocity of radio wave is the same as the velocity of light, which is 3x108 m/s. What is the wavelength of this radio wave?
Answer. The frequency f is 100x106 Hz and the velocity v is 3x108 m/s. From the formula above, the wavelength is
Copyright 2010 by Kai Hock. All rights reserved.
Last updated: 22 May 2011.