Natural Vibrations and Resonance of a String – What Happens When a Periodic Vibration Is Applied to the String?

Frequency Fine-Tuning － ⇔ ＋     Mass Ratio     Spring Constant Ratio 　　Reset 　　Slow　 Non-Erasing

　　　　　　　　　　　　　

Description :
    This page is designed to observe natural vibrations and resonance by applying periodic vibrations to a string.
    Many objects, whose motion is restricted in the vertical direction, are connected by springs. This system is modeled as a string, and periodic vibrations are applied at the left end using a vibration device. The vibrations generate waves that propagate to the right, where they reflect at the fixed right end and reverse direction. Later, at the vibrating left end, a fixed-end reflection occurs (although the left end vibrates slightly, it is almost fixed), and the direction of the wave changes again, propagating back to the right. The waves generated by the vibration device continue to travel back and forth, overlapping with each other. When certain conditions are met, the waves will constructively interfere with each other, causing the string to vibrate more strongly.
    The state in which the string vibrates with large amplitude is called the "resonant" state, and the frequency at that time is called the "resonant frequency." Since both ends of the string are fixed, you will notice that standing waves with nodes at both ends are formed during resonance. Let’s observe this under various conditions.
    In this simulation, you can adjust the amplitude and frequency of the vibration device, as well as the mass of the objects and the spring constant of the string. Changes in the mass of the objects correspond to changes in the string’s linear mass density, while changes in the spring constant affect the string’s tension. Additionally, when resonance occurs, the amplitude of the string’s oscillations increases significantly, so a small damping force is applied. Be cautious: if the mass is set too small and the spring constant too large, the motion of the string becomes excessively intense, leading to significant computational errors and causing the simulation to break down, potentially "breaking" the string. In such cases, please reload the page.

● By moving the blue section at the bottom left of the screen, you can adjust the amplitude of the vibration device between 0 and 20. The initial value is set to 5.
● By moving the red section at the bottom left of the screen, you can adjust the frequency of the vibration device between 0 and 100. The initial value is set to 2, so try gradually increasing the frequency. You can also adjust it using the fine-tuning buttons (- and +) for frequency adjustment. (At 11, it's approximately 1 Hz.) (Hint: Around 16...)
● You can change the mass of the string using the "Mass Ratio" slider. (This corresponds to the linear mass density of the string.)
● You can change the spring constant of the string using the "Spring Constant Ratio" slider. (This corresponds to the tension of the string.)
● You can reset the system to a non-vibrating state by pressing the [Reset] button.
● By checking the [Slow] checkbox, you can slow down the motion, making it easier to observe.
● By checking the [Non-Erase] checkbox, you can observe the motion by layering it over time.

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