Physics 2025 – Lab 1 Standing Waves Partner 1: Partner 2: Section: Partner 3 (if applicable): Standing Waves Purpose: Continuous waves traveling along a string are reflected when they arrive at the (in this case fixed) end of a string. The reflected wave interferes with the incoming wave, resulting in most cases in an erratic and irregular movement of the string. Under particular conditions (coordinated length of the string, wave frequency, linear mass density of the string, and tension of the string) a standing wave pattern emerges and can be observed very clearly. Under these conditions the wavelength of the standing wave (and thus that of the incoming wave) can be measured. The theoretical relationship between the aforementioned parameters is examined and tested for its consistency in a real experimental setting. Apparatus List: PASCO Mechanical Wave Miscellaneous Cables and Clamp and rod driver Adapters Capstone Software Clamp and Pulley White Flexible String Pasco Power Amplifier II Hooked weight set Digital Scale (on shared table) (CI-6552A) Two C-Clamps Pasco 750 Interface Theoretical Background What Are Waves And How Can We Classify Them? In very general terms, a wave is a disturbance that travels from one place to another (for example, a stone thrown into a calm pond creates a disturbance that travels along the surface of the pond, away from the site of impact) . In addition, if the disturbance originates from a continuous source, then a continuous wave is created (for example, a wave created by a running motor of a boat). Furthermore, some waves are created by a continuous source that moves sinusoidally. The wave can then be classified as a harmonic wave. In this experiment we generate sinusoidal waves with the help of a computer, Capstone software, the 750 Pasco interface, and a Pasco amplifier. We will use sinusoidal waves in this experiment 5 Physics 2025 – Lab 1 Standing Waves because a sinusoidal (harmonic) wave has a distinct (single) frequency associated with it. In contrast, rectangular or saw-tooth waves are composed by superposition of many sinusoidal waves of different frequency and amplitude. Use of the harmonic waves therefore reduces greatly the number of parameters in our experiment. In other words, we can clearly identify the frequency of the wave by using harmonic waves. Waves can be classified also by the direction of the disturbance. If the disturbance is perpendicular to the direction in which the wave travels, the wave is called “transverse” (an example would be a wave traveling along a string). If the disturbance is in the same direction as the direction in which the wave travels, the wave is called “longitudinal” (example: Sound). Reflection of Waves, Standing Waves When a wave encounters a discontinuity of the medium in which it travels (sound hitting a wall, the end of a string, the water waves hitting the beach) a certain portion of the wave is reflected backwards. The pattern of the reflected wave resembles closely that of the incoming wave. In cases where the discontinuity forces the waves amplitude to be zero at the location of the discontinuity (e.g., when the end of a string is tied to a fixed place so its end cannot move) the reflected wave is inverted. In the opposite case (e.g., a string with a loose end that can move) the reflected wave is not inverted. In either case, if the wave is continuous, there will be basically two waves traveling simultaneously along the medium: The incoming wave and the reflected wave. The two waves will interfere, which means that their respective time and space dependent displacements will add up. In general, adding the incoming and reflected wave results in an irregular wave pattern. However, if the conditions are right, the interference results in a regular wave pattern along the medium. Such a regular pattern is called a standing wave. The Speed Of A Wave Traveling On A String The speed (v) of a wave on a String is determined by two properties: 1) The tension (T) in the string. 2) The linear mass density of the string. The linear mass density can be calculated from the mass (m) of the string and it’s length (l): . 1. The speed can be calculated with this equation: . 2. In addition, the relationship between the frequency f, wavelength , and speed v of the wave is given by the equation ∙ . 3. 6 Physics 2025 – Lab 1 Standing Waves NNA Standing Wave Patterns On A String With Fixed Ends A standing wave can be characterized by the presence of areas where the string does not move at all (called “nodes”) and areas where the string moves with maximum amplitude (“anti-nodes”). The most simple standing wave thus consists of two nodes and one anti-node. The nodes are at the two sides that are fixed and the anti-node is halfway between the nodes Note: The distance between two nodes corresponds to half a wavelength Suppose, starting from this standing wave pattern, you increase the frequency of the wave slowly, you will notice that first the standing wave pattern will disappear and then, when the frequency is right, another standing wave pattern will emerge. This pattern has two Anti-nodes and three nodes. Again, the distance between two nodes corresponds to half a wavelength. Clearly, the wavelength is reduced for this second standing wave pattern by a factor of 2. Problem 1: [1 pt] Assuming that the speed of the wave has not changed when going from the standing wave pattern with one Anti-node to the one with two Anti-nodes, how would the frequency have to change? Problem 2: [ 1 pt] Make a drawing of the next expected possible standing wave pattern that would occur at the next higher frequency. 7 Physics 2025 – Lab 1 Standing Waves Setup: The mechanical wave driver is positioned underneath a string. The string is attached to a rod on one end of the table and redirected over a pulley at the other end of the table. At the end of the string (on the pulley side) a weight hanger (and weight) is attached that creates the required tension in the string. Make sure that the small plug with the U-shaped groove is attached to the moving rod of the mechanical wave driver. The string should run through the groove of that plug. Thus, the mechanical wave driver only lifts and lowers the string periodically. Never attach the string directly to the wave driver’s arm. This may damage the vibrating membrane within the wave driver by creating a sideways tension on the rod. The wave driver itself should be clamped to the table with two C-clamps. NNAAN 8 Physics 2025 – Lab 1 Standing Waves Electrical Connections: Computer/Capstone Pasco 750 Interface Software Channel A Pasco Power Amplifier II Mechanical Wave Driver Procedure: Activity I - Creation of Standing Waves In this activity you will familiarize yourself with the apparatus and experience a few of the basic properties of standing waves. You should have a white elastic string that is long enough to reach from the rod at one end of the table over the mechanical wave driver and over the pulley at the other end of the table and several inches further down. A special banana plug with a center groove is provided as a guide for the string. That plug is to be positioned in the center rod of the wave driver. 9 Physics 2025 – Lab 1 Standing Waves CAUTION ! Make sure you carefully put the locking mechanism of the wave driver into the locked position (you may need to pull up or push down the shaft slightly to be able to properly lock it) during plugging and unplugging of the banana plug. Otherwise the membrane of the wave driver may be damaged. Don’t forget to unlock that mechanism again afterwards (before starting vibrations). Route the string over the pulley and attach the mass hanger set to it. Position the pulley on the table such that the distance from where the string is attached on the rod next to the wave driver to the top of the pulley is exactly 2.00m. The mass hangers plus attached masses provide the tension in the string. Make sure that the mass does not touch the floor when you hang a mass of 250g on the string (note: make sure you take the mass of the hanger itself into account as well, it is usually 50g). If it does touch the floor, you probably need to attach the mass a bit further up on the string. Problem 3: [ 1 pt] Write down the equation with which you can calculate the tension in the string from the attached mass (M). (Hint: Don’t use Eq. 2 for this! Instead consider the free body diagram of the mass. It is very simple!). T = The mechanical wave driver is driven by the power amplifier, which in turn is controlled by the computer through the Pasco 750 interface. To start the oscillations of the mechanical wave driver, you need to open the file “Standing Waves.cap”. It is available on the elementary lab web page. Please follow the choice of browser indicated on the website. Click on “Capstone Files” and then click on “Standing Waves.cap”. A file dialog opens asking you whether you want to open, save or cancel. Click “Open”. It may take 20 seconds or so for the file to load and start up in the Capstone software. Follow the instructions - given in the Capstone file that you have opened - to control the frequency at which the wave driver operates.