Springs are the first step of learning about waves because they operate with something called simple harmonic motion.
The formula for springs is F= -kx and it is called Hookeís Law
The k is a number specific to the spring in question. The x is the distance from the natural place which the spring holds. The F is the force the spring is giving on the objects attached to either end. This formula is always negative because the spring is always trying to get to a point of zero displacement.
Ok, one more time. If you have a spring, it naturally falls in to an equilibrium position of itsí default length. When the spring is pulled to a longer length than the equilibrium position it tries to contract back to itsí equilibrium position. If the spring is condensed then it tries to push back to itsí equilibrium position.
Not all springs follow Hookeís Law but the ones that do are called ideal or linear springs. But for all the questions idea springs will be used.
The energy of SHM
Another way of showing how springs work is by showing the conservation of energy involved. As the spring bounces it converts KE to EPE. When the spring is either extended or contracted it has EPE equal to .5kx^2 when it is in the center the block has momentum and has the energy of .5mv^2
A simple pendulums is a weight attached to rod or string without mass which swings back and forth from a vertical equilibrium position.
A pendelum uses the concept of conservation of energy for it to continue moving back and forth. As it moves to teh top of it's arc it loses KE and gains PE. As it nears the bottom of it's arc it's potential energy is converted to KE.
Now how does all this relate to waves.
Well in essence the block moving up and down creates a wave. If the block was oscillating up and down it would draw a wave along a sheet of paper which is moving at a constant speed. So the block if actually moving in a wave. This is characterized by itís ability to do a complete cycle (from top to bottom to top) is the exact same amount of time every time the block cycles.
This is called the period (definition: the time which a wave takes to make a complete cycle). Another way of explaining the number of cycles the box goes through per second, called frequency.
A relationship exists between these two terms.
A wave is a disturbance moving through a medium without the medium actually moving.
The most basic example of a wave is if you were a holding a rope anchored at the other end and you started to move you hand up and down at a steady speed. Parts of the rope are just going up and down, they arenít actually moving to the wall and back to your hand. In this way a wave is unique because pressure or energy to be transferred without any matter being moved.
A transverse wave consists of a wave moving up and down similar to a sine wave. A longitudinal wave is a wave which instead of going up and down it compresses the medium it is going through. It would be equivalent of you holding a spring and pushing straight forward and pulling out and then pushing and pulling so that groups of more condensed spring coils would be throughout the spring.
The speed a wave is travelling with varies completely on the way the medium is set up and not the way the wave is. To be more specific it varies on the tension of the string, mass and length and in no way is dependent on the wavelength and the frequency.
Is the "linear mass density" and is m/L of the string
Remember speed is determined by medium and not by the wave. When a wave changes medium it has to change speed. But since all waves no matter what frequency they have must go at the same speed through the same medium, frequency must be related to something to keep the constant speed.
As frequency increases, the wavelength will decrease but it will always be the same speed when in the same medium.
Interference of Waves
Interference between waves occurs when you have two waves going through the save medium and they cross over each other. Ok interference with waves can be both constructive and destructive. AS the names say, constructive is where the waves build on each other to produce larger waves and destructive is where the waves cancel each other out to produce smaller waves.
First Iím going to give you a real example and then Iím going to give you a working analogy. If you have two friends on opposite sides of a long rope and you both jerk the rope up constructive interference will occur in the center of the rope when the two bulges over lap. The best way to think about this is to imagine that when two waves cross their heights along the rope and added together.
Or you can think of it this way. Letís say you have two equal piles of dirt. As you bring them closer and closer they start to hit each other and build off each other. Then they are together and are twice as big as the original two waves. It is like you putting one mound on top of the other and you would have constructive interference.
If you have two friends on opposite sides of a long rope and one of you jerks the rope up while the other jerks the rope down destructive interference will occur in the center of the rope when the two bulges over lap. Think of it as the downward jerk cancels out the upward jerk. Using the previous analogy, it is like having a hole in the ground (the downward jerk) and a pile a dirt (the upward jerk). And when the waves get close to each other they cancel each other out.
A standing wave occurs when you have two ends of a rope attached to something so that when a wave gets to the end of the medium (in this case the rope) it reverses direction but will have the same frequency, amplitude and wavelength. When these two waves are overlapping in just the right way a standing wave will appear. A standing wave can be characterized by the way some points donít move vertically at all and other points move large distances.
This is cause by destructive and constructive interference through out the wave. Where the rope appears to not move at all (called a node) destructive interference is at work. Where the rope moves a lot (called an anti-node) constructive interference is at work.
The distance between two anti-nodes and two nodes is always Ĺ lambda and the distance between an anti-node and a node is always ľ lambda.
A standing wave is produced when the length of the string is proportional to the wavelength of the frequency.
N is equal to 1,2,3Ö.
From this equation a fundamental frequency can be derived by know the speed waves travel through that particular medium and the equation
Once you have the fundamental frequency you can simply times it by n to find all the other frequencies and wavelengths.
Sound waves are produced by a vibration of an object which causes compressions in the air which vibrate your ear drum.
The human ear on average can detect sounds frequencies in between 20Hz and 20000Hz. That is why if you ever look at a set of headphones they will advertise how close they can get to these frequencies.
Number one is an example of a rarefaction or where the air is less compressed then normal. Number two is a compression and an example of where the air is more condensed then normal.
When you have two sound waves within frequencies of each other, they produce a beat frequency when played at the same time. This is a product of constructive and destructive interference because if they have different frequencies they will go from being in sync to out of sync to in sync again. The number of beats per second is equal to the absolute value of the difference of the two frequencies.
In this way you can find how much of a difference between the frequencies exists.
Sound Standing Waves
Standing waves can be set up in pipes, which are open on one end and closed on the other end. If you have a tube with one side vibrating and the other fixed you have the equivalent to having a standing wave inside the pipe. This is because on the fixed end the vibrating acts as a node (the part of the standing wave that isnít moving at all) and the open end acts as an anti-node (the part which moves the most).
If the antinode is at one end of the tube and the node is at the other end of the tube then