A Model of a Traveling Periodic Longitudinal Wave

A MODEL OF A TRAVELING PERIODIC LONGITUDINAL WAVE Vw System System Map Temporal Representations Vw Vw Vp T1 Vp T2 T3 Vp Vp Vp Vp Rest Average Velocity Vw Vp Graphical Representation of T1 Pressure Displacement Vp Vp Vp Vp Mathematical Representation Zero f*λ =v Verbal Representation • A longitudinal wave requires a medium with particles that interact with each other. As a longitudinal waves passes, a vector representation of the movement of a particle in the medium is always in the same direction as the movement of the wave itself or the exact opposite. Unlike transverse waves, it is never perpendicular to the direction of propagation of the wave. • A single particle of the medium moves forward and back relative to the movement of the wave itself. • The movement of one particle affects the movement of its neighbors. In its rest position, when no wave is present, the forces acting on a particle are balanced. • When a wave is present, the graphical representation of the displacement is zero, when the velocity is at a maximum or a minimum. The slope of the velocity graph is zero at maxima and minima. Therefore, the acceleration, and thus the net force, must be zero when the displacement is zero. • Consider that an outside agent (i.e., OA) moves a particle towards another particle; the two particles repel as if a spring between them were compressed. Conversely, if an OA moves a particle away from another particle; the two particles attract as if a spring between them were stretched. C A B C A B • • • • Before After Let the OA move particle A of the medium towards fixed particle C of the medium. The original particle, particle A, repels, pushes on particle C in one direction, and particle C repels, pushes on particle A the other way. Meanwhile, particle B, located on the opposite side of particle A, attracts, pulls on particle A one direction, and particle A attracts, pulls on particle B the other way. 1 C A B FC FB Forces on A by B & C • • • Let the OA release particle A. Particle A will accelerate towards its rest position of zero displacement due to the net force towards B. However, at the rest position, the net force on particle A will be zero due to equal forces exert by particles C and B. This assumes the “spring” on both sides of A are stretched equally. As particle A’s momentum carries it through the rest position, the displacement and force diagrams reverse. C A B FB FC Forces on A by B & C • • • • • Velocity • Notice that the area of closely spaced particles has traveled towards particle B. This is the compression area of the wave. Described below. Particle A slows down due to the net force in a direction opposite to its motion. Particle A’s velocity eventually reaches zero at its maximum displacement as particle C pulls with a greater and greater force. As particles A and B converge, (i.e. the distance between them decreases), the repulsive force between them increases, much like a compressed spring. The name “compression” is given to the area in which the particle spacing is less than normal, and the particle’s velocity is in the same direction as the wave’s velocity. It would be at a maximum when the pressure of the particles around particle A is high, the displacement of particle A is zero, and the velocity of particle A is at a maximum. Consider the force on particle B by particle A. It would be a push in the direction of the wave’s motion. Therefore particle B is pushed to the right of its rest position as particle A reaches its maximum displacement. Look at the displacement graph in the graphical Graphical Representation of T1 representation. Find the maximum positive displacement. The corresponding velocity must be zero as the particle turns around. At that Rest point on the velocity graph, the slope is very negative, which indicates a very large acceleration, as well as net force, in a direction opposite to particle A’s motion so far. Eventually the pushing force on particle A Average exerted by particle B and the pulling force exerted on particle A by particle C result in particle A speeding up in the opposite direction, (i.e. toward particle A’s rest position). Zero Pressure Displacement • • Particle A will accelerate towards its rest position of zero displacement due to the net force towards C. C A B FB FC Forces on A by B & C • • • However, at the rest position, the net force on particle A will be zero due to equal forces by particles C and B as before. The particle will be at its rest position when its speed is maximum in the negative direction, (i.e. opposite to the wave motion.) Particle A accelerates backward, gaining speed in the negative direction until eventually the distance between particle A and particle B becomes so large that particle B’s push becomes a pull much like a compressed spring as it becomes a stretched spring, and particle C’s pull becomes a push much like a stretched spring as it becomes a compressed spring. 21 C A B FC FB Forces on A by B & C • • Velocity • The particles are separated the greatest amount when particle A reaches its maximum negative velocity. The name “rarefaction” is given to the area in which the particle spacing is more than normal, and the particle’s velocity is in the opposite direction as the wave’s velocity. It would be at a optimum when the pressure of the particles around particle A is low, the displacement of particle A is zero, and the velocity of particle A is at a maximum in the “negative” direction, (i.e., opposite the direction of the wave). Particle A’s momentum carries it through the rest position, as the displacement and force diagrams reverse. Look at the displacement graph in the graphical Graphical Representation of T1 representations. As particle A reaches its maximum negative displacement, the velocity must be zero because the particle changes Rest direction there. The slope of the velocity graph is its most positive there also, and therefore the acceleration and net force are the greatest in the forward direction, towards particle B, when particle A is displaced the most in the backward Average direction, towards particle C. As particle A moves away from particle B and towards particle C, the pull from particle B and the push from particle C increase. Particle A Zero slows to a zero velocity, and reaches its maximum negative displacement due to the net positive force, (i.e., opposite to the direction of particle A’s motion). Particle B’s force on particle A is equal and opposite to particle A’s force on particle B. The same is true of the particle A & C pair. Particle A pushes particle C away from its rest position and pulls particle B towards its maximum negative displacement. Eventually the forces on particle A combine to accelerate it forward to its original position where it has its greatest speed in the forward direction once again. If the cycle repeats, each particle has its greatest speeds as it passes through its original position first in one direction and then the other. The oscillation of the particle moves from one particle to the next only because they interact, (i.e., apply forces on each other). As a longitudinal wave moves from particle to particle through the medium, each particle is displaced forward and then backward to its rest position. As a particle is displaced, it alternately slows down and then speeds up due to an unbalanced force acting upon it as the neighboring particles apply unequal forces. At the point of maximum displacement, the particle has maximum net force on it, a zero velocity, and the surrounding particles are either the closest they will get (i.e., high pressure) if the particle is moving in the direction of the wave, or the farthest apart they will get (i.e., lowest pressure) if the particle is moving in the opposite direction of the wave. A longitudinal wave always has its displacement 90 degrees out of phase with its compression component. Pressure Displacement • • • • • • • • MODEL OF A SOUND WAVE IN A CLOSED TUBE Using the model of LONGITUDINAL GAS WAVES (SOUND) 31 OF THE MECHANICAL WAVES MODEL Vw BEFORE System Geometric Represention closed end Displacement Temporal Vp Representations Reflection of sound Vp AFTER open end Graphical Representation Displacement (L) Velocity 1/2λ Pressure 1/4λ Mathematical Representation λ= 4L f* λ= v 41 Temporal Representation (Continued) (Note: Toward the right is positive) Displacement a b: Velocity a c: Acceleration Graphical Representation Open End Net Force a: d: (L) e b c d f 1/2λ a 1/4λ d b a b b c c c e d d f: g: 51 g f e e Force Diagram of End Particle a b c d e f g e: g f f g g Verbal Representation Resonance in a closed tube (with a sound wave) 1. When this wave enters a closed end tube the compression portion of the wave strikes the closed end of the tube and rebounds (not inverting like the transverse wave.) The closed end of a tube is a pressure antinode where there is maximum pressure differential (maximum to minimum.) 2. The closed end has a displacement node where the particles can not move. There is displacement node at the closed end because the particle must strike the rigid end and have no displacement variation. The closed end of the tube is a displacement node where there is no displacement variation. 3. The closed end particle velocity is zero. If displacement is zero the velocity is zero. There is a velocity node at the closed end because the particle must strike the rigid end and have no velocity. 4. The closed end of the tube is an area of maximum pressure ( + or - ) where the reflection of either a compression or rarefaction must occur in order for a standing wave to form. 5. When the reflected pressure wave approaches the opening of the tube the atmosphere it is allowed to freely leave. The pressure at the opening of the tube is at equilibrium and is a pressure node. Eventually the pressure in the tube drops and a compression of particles once again travel down the tube. This new compression is 180 degrees out of phase with the exiting rarefaction part of the wave. The open end has a pressure node where the wave is open to the air. It is the place where a compression meets a rarefaction. 6. When that wave exits the open end of the tube there is displacement antinode. The particle is free to move and shows a maximum displacement variation. 7. When that wave exits the open end of the tube there is velocity antinode because the particle is free to depart the tube. Particles are also rushing in due to the drop in pressure along the tube. The result is a maximum variation in particle velocity. 8. A rarefaction could enter the opening to begin the process. If it reflects back on its own compression, the velocities would add to give an inward velocity. 61