Directional discontinuities are frequently encountered in the solar wind. This study utilizes the possibility of simultaneous four‐point measurements that the Cluster satellites provide for an investigation of the waves near one such interplanetary discontinuity event. In particular, the k‐filtering technique has for the first time been applied for the wave characterization in terms of frequency, wave vector, wave power, and polarization at different distances from a discontinuity. The advantages of the k‐filtering method are that these parameters can be transformed into the plasma frame of reference to allow comparison with theory and that it is possible to detect more than one wave mode at each frequency in the spacecraft frame of reference. The discontinuity event in this study was chosen because its wave activity had unusually high power and because it also contains signatures of large‐scale magnetic reconnection, such as a reconnection exhaust. Two wave modes are found: one which has the features of a shear Alfvén wave with propagation direction ordered by the background magnetic field direction and one which behaves like a compressional Alfvén wave and is ordered by the discontinuity normal. Both wave modes become weaker farther away from the discontinuity, but the compressional Alfvén mode is more suppressed. The shear mode dominates for long wavelengths at all distances from the discontinuity. The wave field is also found to be asymmetric on the two sides of the event.