ChemPhys 173/273

Unit 3: Wave Basics

Problem Set A

Overview:

Problem Set A targets your conceptual understanding of waves and your ability to mathematically analyze wave patterns. There are 20 problems in the set with more than half of them involving routine, single-step mathematical manipulations. To be successful at solving these 20 problems, there are a variety of conceptual understandings and mathematical skills which you will have to master. These concepts and skills include but are not limited to:

• The use of the wave equation.

There is a mathematical relationship between the speed or velocity (v) of a wave and the frequency (f) and wavelength () of the wave. That relationship is expressed by the wave equation

v = f •

Knowledge of two of the three quantities allows one to calculate the third quantity.

• The use of the frequency-period relationship.

There is a mathematical relationship between the frequency of the wave and the period of the wave as denoted by the following relationship:

f = 1 / T

Knowing frequency allows one to automatically calculate the period, and vice versa. It is important to recognize that frequency is a rate quantity which identifies the number of times a periodic event happens per unit of time. Frequency units are always waves/second, vibrations/second, oscillations/second, cycles/second or something comparable. This is also known as a Hertz (Hz). Period is the time it takes for a periodic event to complete a single cycle. Period units are simply seconds (or minutes, or hours, or some time unit).

• The definition of wave speed as a distance to time ratio

The speed (or velocity) of a wave is defined as the distance a crest on the wave travels per unit of time. Like the speed of a runner or a car, wave speed (v) is simply the ratio of the distance (d) traveled per time of travel (t) as shown by the following equation.

v = d / t

• The analysis of a standing wave pattern

When a wave is introduced into a medium such as a rope or a snakey, it travels the length of the medium and then reflects back upon reaching the end of the medium. At certain frequencies, the reflected portion of the wave meets up with the original wave to create a pattern known as a standing wave pattern. In a standing wave pattern there are points along the medium which appear as if they are always standing still. These points are known as nodes and are easily remembered as the points of no displacement. Separating the nodes are anti-nodes: points of maximum positive and negative displacement. In such standing wave patterns, there is a unique half-number relationship between the length of the medium and the wavelength of the waves which have established the pattern seen. These relationships are shown below for the standing wave patterns having one node (first harmonic), two nodes (second harmonic) and three nodes (third harmonic).

From the pattern and knowledge of the length of the medium, the wavelength can be determined using the above equations (or vice versa).

• A solid understanding of wave definitions and concepts

Physics problems are more than mere exercises in the mathematical manipulations of algebra formulas. Make no mistake about it, the solution to many physics problems requires an ability to mathematically manipulate an equation in order to solve for an unknown variable. However, many physics problems require more than mathematics. In fact, in many instances a student exercising good mathematical skills misses a problem because of a conceptual misunderstanding. In many of the problems in this set, you will need to use your understanding of wavelength, frequency, period and wave speed to appropriately interpret the verbal information in the problem statement. For instance, you will have to read a phrase such as "the oscillator that generates the wave completes 46 vibrations in 53.5 s" and use conceptual understandings to appropriately extract the numerical information from the problem and associate it with variables of the equations. Furthermore, there are many instances in which extraneous numerical values are stated in the problem description; such values do not need to be used in the solution. This extraneous information will only be a distraction to students who treat physics problems as mere mathematical exercises with little connection to physics concepts.

Physics is about conceptual ideas and relationships; and problem sets test your conceptual understanding of these relationships. If you treat this problem set as a mere exercise in the algebraic manipulation of physics equations, then you are likely to become frustrated quickly. As you proceed through this problem set, be concepts-minded. Never strip physics of its conceptual meaning.

• Attention to units

Some of the problems involve the expression of given information in non-conventional units and will require unit conversions. As is the case in any problem set, it is critical to pay attention to the units associated with given and calculated quantities and to enter answers in the unit which is requested within the problem statement.

The following pages from The Physics Classroom tutorial may serve to be useful in assisting you in the understanding of waves and the mathematical analysis of their patterns.

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View Sample Problem Set.

 Problem Description Audio Link 1 Routine calculation of period. 2 Routine calculation of wave speed. 3 Routine calculation of wavelength. 4 Referring to the previous problem; routine calculation of wavelength. 5 Multiple step problem for calculating the wavelength; requires attention to units. 6 Routine calculation of amplitude. 7 Referring to the previous problem; determination of wavelength from a pattern. 8 Referring to the previous problem; routine calculation of period. 9 Referring to the previous problem; routine calculation of wave speed (requires correct answer on #7). 10 Multiple step problem for calculating the wave speed; requires attention to units. 11 Use of the definition of speed to calculate a time of travel; requires attention to units. 12 An echo problem involving the use of the definition of speed to determine a distance of separation. 13 Multiple step problem for calculating the wave speed. 14 Referring to the previous problem; routine calculation of amplitude. 15 Multiple step problem for calculating the frequency. 16 Analysis of a standing wave pattern to determine the wavelength of a wave. 17 Referring to the previous problem; routine calculation of wave speed (requires correct answer on #16). 18 Multiple step problem involving the analysis of a standing wave pattern and the determination of the frequency. 19 Multiple step problem involving the analysis of a standing wave pattern and the determination of the number of wave cycles in a specified amount of time. 20 Multiple step problem involving the analysis of a standing wave pattern and the determination of the wave speed.

Audio Help for Problem: 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15 || 16 || 17 || 18 || 19 || 20

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