In this module, we will analyze waves and oscillatory motion. All waves transfer energy from one location to another. Imagine a rope held taut by two students at either end. When one student begins to shake the rope up and down, a disturbance or wave pulse is generated in the rope, and carries energy toward the student at the opposite end. This is a good example of a mechanical wave. Mechanical waves require a medium, in this case the rope, through which to travel. Light, which is more of a AP Physics 2 topic is not a mechanical wave but an electromagnetic wave and does not require a medium. Notice that in this example, with the rope and the students, the energy of wave moved to the right. The rope, our medium, did not move to the right with the wave energy. In fact, if you watch one section of the rope during the wave motion, you will see it move up and down, not left and right. Since the medium travels perpendicularly to the wave motion, we call this a transverse wave. Some waves, such as sound, are longitudinal. The particles vibrate in a direction parallel to the motion of the wave. The vibration of your vocal chords against the air, compresses and expresses the medium, as the wave travels away from its source. >> There are lots of ways to represent and describe a wave in motion. Suppose you are looking at a transverse wave on a string, as seen here. A wave length is the distance along a wave between two identical points, the distance it takes for the wave to begin repeating itself. This can be from crest to crest, trough to trough or any point in between. The frequency of a wave is defined as, the number of complete cycles of a wave that occurs per unit time. We represent oscillations per second as hertz. The inverse of frequency is called a period of a wave. It represents the time it takes for one oscillation or revolution to occur and it is measured in seconds. The wave equation seen here, as v equals frequency multiplied by wavelength, can help us calculate how quickly a wave crest moves along a wave. This is not the speed of the individual particles in the medium, but is the overall speed of the wave crests. >> So in this question, they tell us that the water wave is seen to pass the edge of the dock every two seconds. This gives us the amount of time for that one revolution, so this is providing us with the period. So we have T=2 seconds. Then they tell us that the observer measures a distance between wave crest to be three meters. So, going from a crest to a crest, that represents the wavelength, three meters. They want to know how fast is this wave moving. So, I'm going to use my equation, I'll call this part A. I'm going to use my equation v = frequency multiplied by wavelength. Well, remember that, frequency and period are related as inverses of each other. So I can go ahead and write this out as v = 1 over T multiplied by wavelength, which is 1 over 2 multiplied by the wavelength, 3. And this give me V equals 1.5 meters per second. So that's part A of this. The next part says, suppose if the amplitude of the wave is 0.25 meters. Calculate the average vertical velocity of the water, located at the edge of the dock. So, say right over here, I've got the wave that's showing up over here, and this is my equilibrium, the water level. So, if this wave travels from here, from this point right over here, if this point travels down 0.25 and then again another 0.25, back up 0.25, and back up over here 0.25, well then the total distance that this point has traveled is, 0.251234 multiplied by 4. So when I try to solve for this speed over here, this average velocity, I'll have V equals X over T. Which is going to be 0.25 multiplied by four, which is 1 divided by the time that it took for that one revolution, just two seconds. And so I have, .5m/s as my answer. >> Reflection occurs when a wave encounters a new medium. Some of the waves energy bounces off the new medium, while some is transmitted into the new medium. The law of reflection tells us that the angle of the incident wave is equal to the angle of the reflected wave, when measured to the normal. For example, if a wave is at 30 degrees to the normal of a surface, it will reflect at an angle of 30 degrees to the same normal. When waves reflect off of a new medium, it doesn't just change direction. The wave might flip upside-down, what we call inversion. Imagine a rope tied to a wall. If a wave pulse is sent down the rope towards the wall, it will hit this fixed boundary. Since the rope is displaced upward, it will pull up on the wall. Because of Newton's 3rd Law, the wall will then pull down on the rope with an equal force. Due to this force on the rope, the wave will flip upside-down when it is reflected back. We call this an inverted reflection. If the rope is instead attached to a ring that is free to move up and down, the wave does not get inverted upon reflection. Since it is free to move, the wall can not pull down on the rope and the wave returns upright. A fixed end and free end boundary, approximate what occurs when a wave travels from one medium into another. If a wave is moving from a light string into a heavy string, this means the wave reflects back inverted, as if it hit a fixed boundary. The opposite is true if the wave travels from a heavy string to a light string. The reflected wave will be upright. >> Interference is what occurs when two waves interact. When two waves overlap, the result and displacement is the sum of their individual displacements. Suppose a wave pulse and a string is traveling to the right with amplitude three centimeters, when it encounters a wave pulse traveling left with an amplitude of 2 cm. The resulting pulse will be 5 cm when they overlap for just an instant. Afterwards, the two waves continue in their motion unaffected, as if nothing had occurred. With the 3 cm pulse moving to the right and the 2 cm pulse moving to the left. When the resultant wave has an amplitude that is larger than either of its components, like in this example, we call this constructive interference. When this is not true, and the waves have opposite displacements, the result is destructive interference. When certain criteria are met, a wave can exist that does not appear to move. We call this kind of wave a standing wave. Look at our two students holding a rope taut, again. If the students send waves at just the right frequency, the resultant wave will have a larger amplitude and appear stationary. Locations of destructive interference are called nodes. And locations of constructive interference are called anti-nodes. For a standing wave to exist, the length of the medium must be a multiple of half wavelengths. We call the frequencies at which standing wavelengths occur, natural or resonant frequencies.
