Read the axis label before the curve

Both of these look like a sine curve, and they answer different questions.

1. Displacement against POSITION. A snapshot of the whole wave at one instant. Its repeat distance is the wavelength $\lambda$, in metres.
2. Displacement against TIME. A record of one particle, over time. Its repeat interval is the period $T$, in seconds.

Grabbing whatever number the horizontal axis offers and calling it the period is how everything computed afterwards goes wrong, with units that quietly do not match. So read the axis label first and say out loud what the graph shows: metres across the bottom means the repeat length is $\lambda$; seconds means it is $T$.

Getting $v = f\lambda$ from graphs usually takes one of each kind, so confirm that $\lambda$ came off a position axis and $T$ came off a time axis before multiplying anything.

$$v = f\lambda, \qquad f = \frac{1}{T}.$$

Two standard situations, and they pin different quantities:

1. One medium, source changed. The string's tension and mass per length fix $v$. Double the vibrator's frequency and $\lambda$ halves. The speed does not move.
2. Crossing into a new medium. The source keeps driving the boundary at its own rate, so $f$ is what carries across. The new medium sets a new $v$, and $\lambda$ adjusts to match.

Reading $v = f\lambda$ as three free variables and holding the wrong one fixed is the error. Usually $\lambda$ is kept and $v$ is allowed to ride along with $f$, which claims that a higher note travels faster than a lower one through the same air. Run backwards, the same slip says a wave slowed down in a new medium because its frequency dropped.

Amplitude is the maximum displacement from equilibrium, and it is independent of the period and the frequency. Loudness tracks amplitude; pitch tracks frequency.

So a curve drawn twice as tall with the same repeat length is the same note played louder. It is not a higher note, and its wavelength has not changed.

The two are read off different axes of the same graph:

1. Height above equilibrium (vertical) gives the amplitude.
2. Horizontal repeat gives the period or the wavelength, depending on which graph you are looking at.

Part of why they feel like one quantity: wave energy rises with both amplitude and frequency, so a louder note and a higher note both deliver more energy. That is a real connection, and it is not an identification.

The vocabulary is worth pinning to the physics, because the everyday words are where the confusion starts.

1. Pitch is how we perceive frequency. Higher $f$, higher note.
2. Loudness is how we perceive amplitude. Bigger displacement, louder.
3. Timbre, which makes a violin and a flute at the same pitch sound different, comes from the mix of harmonics present, which is topic 14.6.

None of the three is the wave speed. Sound of every pitch and every loudness travels through room-temperature air at about $343$ m/s, which is why an orchestra arrives in time with itself from the back of a hall.