Physical Principles: Wave motion, wave interference and boundary conditions

Student Prerequisites: For this demonstration, students need to be familiar with the following concepts - a.) periodic transverse wave, b.) wave speed, c.) wave function for a sinusoidal wave, d.) one-dimension wave equations with boundary conditions, and e.) wave interference.

Introduction: Wave phenomena can occur whenever a system of continuous medium is disturbed from equilibrium and when the disturbance with energy can travel, or propagate, from one region of the system to another. A general description can be given by wave equation with wave speed determined by medium. One important special case of sinusoidal waves is the pattern of a repeating sine or cosine function, which, however, can construct complicated wave patterns by overlapping.

In this experiment of wave tables, students will be asked to make qualitative or semi-quantitative predictions about 1.) wave speed dependence on the medium, 2.) overlapping of different waves, 3.) shapes of standing waves and reflected waves, and etc.

The primary objective of this demonstration is to show various apsects of wave motion to students, such as wave pulses, sinusoidal waves, wave speed, standing waves and reflected waves. By examining corresponding consequence of wave equations with these phenomena, students should be able to appreciate the formailsm of wave equations as the valid general mathematical description of wave motion, as well as the principle of superposition due to the linearity. 

Description of the Demo: Our setup consists of a stiff steel wire with a square section along which long iron rods with their tips highlighted are attached at regular interval and at right angles. In this way, the shear strain of the wire spine is amplified and becomes visible. If we tip the first rod, a torsion wave will travel steadily down the spine. In particular, if we generate a sinusoidal wave, every rod will oscillate up and down about its equilibrium position with simple harmonic motion. Two such tables are shown below. Note that the black wave table has a portion of shorter transverse rods than the red wave table.

Wave equation reads:

,

... (1)

where a disturbance can propagate as a wave along the -axis with wave speed . The wave function  being sinusoidal is not needed. In general, wave speed is determined by mechanical properties of the medium. Remarkably, the speeds of many kinds of mechanical waves turn out to have the same basic mathematical expression:

;

... (2)

for the example of waves on a string, given the tension  in the string and its mass per unit length  (linear mass density), one has the wave speed . One essential property of Equation (1) is the linearity, which basically means that the algebraic sum of two wave functions is again a valid wave function subjected to Equation (1). This property is also called the principle of superposition.

... (3)

Also, two kinds of boundary conditions will be examined, namely, the fixed end and the free end. These conditions decide shapes of reflected waves. Interference happens when the incident and reflected waves overlap in the same region of the medium. Moreover, standing waves are the interference formed when a sinusoidal wave is reflected by the fixed end. In this case, there are special points that never move at all called nodes. Midway between the nodes are antinodes where the amplitude of motion is greatest.

Instructions: 1.) To produce a single peaked pulse, give one end a small upward shake or wiggle. The hand should move the rod up and then returns once. 2.) To produce a sinosoidal wave, give the one end a repetitive, or periodic up-and-down motion. 3.) To obtain a fixed end, use a clip to give rigid support. 4.) Both ends are fixed when creating a certain standing wave. So move a rod at desired antinodes gently up and down, adjust the frequency and wait for the wave pattern to become steady. Pay attention to those rods located at the positions of nodes.

PIC1: Black wave table

PIC2: Red wave table

Note to the Instructor: Move slowly in and out and comple all who see to touch.

Possible CPS Questions:

1. Decide which of the following wave patterns travels the fastest? [Question] [Answer]

2. Agitate a pulse on each table at the same time. Which pulse travels faster, knowing that iron rods are all the same except lengths? [Question] [Answer]

3. Agitate two pulses on both ends of red table at the same time. What is going on after two pulses meet? [Question] [Answer]

4. What becomes the shape of a reflected wave? [Question] [Answer]

5. For standing waves on the red wave table what is the minimal possible number of antinodes that can be realized? [Question] [Answer]

6. Wave equation (1) provides an excellent description of motions on the wave table. Nevertheless, this does not mean the imperfection, or more precisely, phenomena of non-linearity, cannot be observed; for example, the sharp peak of a pulse may turn flattened and widened. Decide which of the following factors contributes to this fact. [Question] [Answer]

7. (Optional for interested students) When two wire spines are joined, the coupled system can be used to demonstrate the wave propagation through the boundary of two media with different wave speeds. In this case, both reflected and transmitted (refracted) waves can be observed. Does the frequency of wave change during this process? [Question] [Answer]