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Wave speed is determined by the properties of the medium, not by the frequency or amplitude of the wave. For a string, v = √(T/μ) where T is tension and μ is linear mass density. For sound in a gas, speed increases with temperature. When a wave crosses from one medium into another, frequency is preserved but wavelength and speed both change according to v = fλ.
Perform experiments varying string tension and mass density, measuring how wave speed changes. Then solve problems where a wave crosses a boundary and students must find the new wavelength given the new speed.
From your study of wave properties, you know that a wave is characterized by frequency, wavelength, amplitude, and speed, all linked by v = fλ. But where does the speed come from? The fundamental insight is that wave speed belongs to the medium, not to the wave itself. The wave is a disturbance propagating through a material; how fast that disturbance travels is set by the material's physical properties. You, as the source, control the frequency — but you have no direct control over how fast the wave moves once it enters the medium.
The formula v = √(T/μ) for a transverse wave on a string captures this physically. Tension T is the restoring force: it pulls a displaced segment of string back toward its equilibrium position. Greater tension means stronger restoring force, so the medium snaps back faster, and the wave propagates more quickly. Linear mass density μ (mass per unit length) is inertia: a heavier string resists being set into motion. The wave must accelerate each segment of string as it passes through; more mass means more sluggish response, so the wave travels more slowly. Speed is the outcome of these two competing factors — the balance between how strongly the medium is pulled back and how reluctantly it moves. Sound waves in air follow an analogous competition between elasticity (the medium's springiness, related to pressure) and density; warmer air is less dense, which is why the speed of sound increases with temperature.
The most important consequence of medium-determined speed is what happens at a boundary where one medium meets another. When a wave crosses from medium 1 (wave speed v₁) into medium 2 (wave speed v₂), frequency cannot change. Frequency is the rate at which wave cycles arrive, which is set by the source that launched the wave — nothing at the boundary can change how often cycles are produced. But if frequency is fixed and speed changes, then wavelength must change to preserve v = fλ: the new wavelength is λ₂ = v₂/f = λ₂ = λ₁ · (v₂/v₁). A wave entering a slower medium shortens its wavelength; a wave entering a faster medium lengthens it. This wavelength adjustment at boundaries — happening at every cycle, simultaneously across the entire wavefront — causes the wave direction to bend when the wave hits the boundary at an angle. That bending is refraction, which you will study in optics, and it follows directly from the same principle: medium controls speed, source controls frequency, and wavelength adjusts accordingly.