wave-1Wave MotionA wave is a self-propagating disturbance in a medium. Waves carry energy, momentum, information, but not matter.Examples:1) Sound waves (pressure waves) in air (or in any gas or solid or liquid)2) Waves on a stretched string3) Waves on the surface of water4) "The Wave" at the ballpark stadium. The medium is the people.5) Electromagnetic waves (light) – this is the only kind of wave which does not require a medium. EM waves can travel in vacuum. This was quite a surprise to 19th century physicists who had trouble imaging a wave without a medium. The thing that is "waving" in an EM wave isan electromagnetic field which generates itself as it propagates. In a sense, the EM waves "rolls out it own carpet", creating its own medium as it moves forward. More on EM waves next semester.We can use a wave to send a signal (information) without sending any matter. Imagine a long line of people holding hands. We can send a signal down the line by hand-squeezing (a disturbance in the people medium), and yet no one moves.Traveling Waves can be categorized as:Traveling waves can also be categorized as:transverse: displacement of the medium is perpendicular (transverse) to the direction of the wave velocity (like a wave on water or a string or drumhead).orlongitudinal: displacement of the medium is parallel to the direction of the wave velocity (like sound wave or a slinky that has been push-pulled).Logitudinal Wave:wave speed vsinusoidal: impulse: vor wave velocity vdisplacementwave-2Mathematical description of traveling sinusoidal wavesSinusoidal waves have a wavelength and a frequency f = 1/T . (Impulse waves have neither.)y = y (x, t)y = displacementy = displacement from the equilibrium (no wave) positionSnapshot in time: freeze time at, say, t = 0xy(x, t ) Ysin� ���= = p����� �l0 2Now freeze position, watch wave go by at position x = 0:ty(x , t) YsinT� ���= = p����� �0 2Wave traveling to the right: x ty(x, t) Y sinT� �� ���� �= p -����� �� �l� �2vxyxy+Y–Yty+Y–Ywave-3Notice: When position x changes by distance or time t changes by period T, the sine function goes through one complete cycle. When x increases by one AND t increases by one T, then thesine function stays at the same phase; we are then riding along with the wave.The argument of the sin functionx tT� �� ���� �p -����� �� �l� �2 is called the phase. A point on the traveling wave (traveling along with the wave) corresponds to a particular value of the phasex tT� �� ���� �p -����� �� �l� �2. As t increases, x must increase in order to keep x tT� �� ���� �p -����� �� �l� �2 a constant value, hence a point of constant phase corresponds to a point moving to the right (increasing x).Could also have a wave traveling to the left: x ty(x, t) Y sinT� �� ���� �= p +����� �� �l� �2We could have used cosine instead of sine for the form of the wave. The only difference betweensin and cos here is where we put the zero of time.Wave speed is distance 1 wavelengthvtime time for 1 to go by Tl= = =lv fTl= = l ( since frequency f = 1/T )Another way to see that our formula for the wave y = y(x,t) corresponds to a wave moving right with speed v =/T is to rewrite the formula like so:( )x ty(x, t) Y sin Y sin x t Y sin x v tT T� � � �� � � � � �p l p� �� �� � � �� �= p - = - = -� �� �� �� �� � � �� �� � � �l l l� �� � � �2 22A point traveling with the wave is a point with (x – v t) = constant or x = vt + const. This is the equation for a point moving right with speed v (graph of x vs t has slope x/t = v) xy+Y–Ywave-4Generally, the wave speed v is a constant, independent of and T. The wave speed v depends on the properties of the medium, not on the properties of the wave. Examples: medium = string, properties = tension, mass per length medium = air, properties = temperature, mass per molecule, etcv = f = constant increases as f decreases, decreases as f increaseswave-5Interference of waves Superposition Principle: If two or more waves are present in the same place, at the same time, the total wave is the sum of the individual waves: ytot(x,t) = y1(x,t) + y2(x,t). You get constructive or destructive interference depending on whether y1 and y2 add (both have same sign) or cancel (opposite signs).Waves carry energy. For a string wave, the energy is both KE (string moves as wave passes) and PE (originally straight string must stretch a little as wave passes – elastic PE similar to (1/2)kx2 ).At time 3 above, where is the energy? Answer: in the KE. At time 3, the string is moving, while instantaneously flat.Standing WavesTwo sinusoidal traveling waves of thesame (and therefore the same f = v/ ) andthe same amplitude, traveling in oppositedirections, overlapping in the same region ofspace, make a standing wave.If the ends of a string of length L are fixed, asin a stringed musical instrument, then standing waves are only possible at certain resonant frequencies given by the condition:L n2l= �, n = integer, L = fixed total lengthvtime 1)vtime 2)time 3)time 4)mx nodesanti-nodesends fixedwave-6SoundSound is a pressure wave in air. It is a longitudinal wave. force Fpressure = , parea A=Air consists mostly of oxygen and nitrogen molecules. At room temperature, the molecules have thermal energy and are moving around rapidly (speed 400 m/s), colliding with each other, and with every exposed surface. The pounding of the air molecules on a surface, like the pitter-pat ofrain on the roof, add up to a large force per area.At sea level, atmospheric pressure is 1 atm = 14.7 psi = 1.01 105 N / m2 (psi = pound per square inch)This is a big pressure! We are not ordinarily aware of thisbig pressure because the air pushes on us equally from allsides (even from our insides due to the air in our lungs).The big forces on us from all sides cancel out and there isnot net force on us.Ordinary sounds are caused by relatively small changes in the air pressure at our ear drum. A pressure change of a few parts per million is enough to wiggle the ear drum and make the listenerperceive a loud sound (the ear is sensitive!). When we here a
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