In a stretched string, the particles oscillate vertically while the crest moves to the right. The time taken for the crest to travel 2.0 m is compared at different tensions.
Adjust conditions:
The sliders adjust the values
張力100 N→400 Nで、山が2.0 m進む時間は何倍か？
粒子間隔0.1 mの列では、山が2.0 m進む時間はどうなるか？
The equations explain the phenomenon
12 masses are connected by springsm d²x₁dt²=−k x₁+k (x₂ − x₁)……
2The motion is decomposed into 2 modesω₁=√(km)，ω₂=√(3km)……
3Many particles are connected in a line by springsm d²yjdt²=k (yj+1 − 2yj + yj−1)……
4The force on a small segment of a stretched string is determinedρa d²yjdt²≒(Sa)(yj+1 − 2yj + yj−1)……
5When the spacing a is small compared with the wavelength, the difference between the slopes on either side is expressed using a second derivativeyj+1 − 2yj+yj−1≒a² ∂²y∂x²……
6Substitution into the equation of motion gives the wave equationIn the continuum limit∂²y∂t²=c² ∂²y∂x²……
7The wave speed on a string is determinedc=√(Sρ)……
8A solution that travels without changing shape is obtainedy=f(x − c t)……
9A sinusoidal wave is expressed mathematicallyy=A sin 2π(tT − xλ)……
10The direction of oscillation changes∂²u∂t²=c² ∂²u∂x²……
11A sinusoidal wave is substituted while keeping the spacing a finitev=c×sin(πa/λ)πa/λ……
What this chapter has shown
In this string, the difference between the slopes on either side produces a vertical net force on each segment. The segments oscillate vertically with timing that varies with position, and the displacement pattern travels to the right. The wave speed is c = √(S/ρ). For the same linear density, increasing the tension by a factor of 4 doubles the speed and reduces the time to travel the same distance to 0.50 times its original value.
For the same tension and average linear density, a particle spacing of 0.1 m and wavelength of 0.4 m give a travel time 1.111 times that for a continuous string.
In everyday lifeIncreasing the tension of the same guitar string while keeping its vibrating length fixed increases the wave speed and raises the pitch.
Detailed summary
We connected particles with springs and wrote their equations of motion, 1 particle at a time. For a stretched string, the vertical components of tension give an equation of the same form. As the spacing between particles decreases, this equation becomes the wave equation. In this transverse wave, the particles oscillate up and down while the displacement profile travels to the right (the wave also carries energy). The wave speed on a string is c = √(S/ρ), so increasing the tension by a factor of 4 increases the speed by only a factor of 2. When the particle spacing is no longer negligible compared with the wavelength, shorter waves travel more slowly.